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The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.
Program Details
Preface
1 Fourier Series
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Index
Preface
1 Fourier Series
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bessel Functions Jn(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
The Eigenfunctions Pn(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Index
Shipping Options
- Standard
- Next day air
- 2nd day air
- 3rd day air
Orders within the United States are shipped via Fedex or UPS Ground.
For shipments to locations outside of the U.S., only standard shipping is available.
All shipping options assumes the product is available and that it will take 24 to 48 hours to process your order prior to shipping.
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You will be taken to our partner Chegg.com to complete your transaction.
After completing your transaction, you can access your course using the section url supplied by your instructor.