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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 *J*_{n}(*x*)

_{n}

#### Solutions When *v* = 0,1,2,...

#### Recurrence Relations

#### Bessel's Integral Form

#### Some Consequences of the Integral Forms

#### The Zeros of *J*_{n}(*x*)

_{n}

#### 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 *P*_{n}(cos *θ*)

_{n}

#### 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

### 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 *J*_{n}(*x*)

_{n}

#### Solutions When *v* = 0,1,2,...

#### Recurrence Relations

#### Bessel's Integral Form

#### Some Consequences of the Integral Forms

#### The Zeros of *J*_{n}(*x*)

_{n}

#### 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 *P*_{n}(cos *θ*)

_{n}

#### 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

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