Skip to main content

Humanities, Social Science and Language

Digital Products

Course managementreporting, and student learning tools backed by great support.

McGraw Hill GO
Greenlight learning with the new eBook+

Personalize learning and assessment

ALEKS® Placement, Preparation, and Learning
Achieve accurate math placement

Ignite mastery of MS Office and IT skills

McGraw Hill eBook & ReadAnywhere App
Get learning that fits anytime, anywhere

Sharpen: Study App
A reliable study app for students

Virtual Labs
Flexible, realistic science simulations


Inclusive Access
Reduce costs and increase success

LMS Integration
Log in and sync up

Math Placement
Achieve accurate math placement

Content Collections powered by Create®
Curate and deliver your ideal content

Custom Courseware Solutions
Teach your course your way

Professional Services
Collaborate to optimize outcomes

Remote Proctoring
Validate online exams even offsite

Institutional Solutions
Increase engagement, lower costs, and improve access for your students


General Help & Support Info
Customer Service & Tech Support contact information

Online Technical Support Center
FAQs, articles, chat, email or phone support

Support At Every Step
Instructor tools, training and resources for ALEKS, Connect & SIMnet

Instructor Sample Requests
Get step by step instructions for requesting an evaluation, exam, or desk copy

Platform System Check
System status in real time

Real and Complex Analysis
Real and Complex Analysis

Real and Complex Analysis, 3rd Edition

ISBN10: 0070542341 | ISBN13: 9780070542341
By Walter Rudin
© 1987

Format Options:

* 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.

Instructor Information

Quick Actions (Only for Validated Instructor Accounts):

This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.


Prologue: The Exponential Function

Chapter 1: Abstract Integration

Set-theoretic notations and terminology

The concept of measurability

Simple functions

Elementary properties of measures

Arithmetic in [0, ∞]

Integration of positive functions

Integration of complex functions

The role played by sets of measure zero


Chapter 2: Positive Borel Measures

Vector spaces

Topological preliminaries

The Riesz representation theorem

Regularity properties of Borel measures

Lebesgue measure

Continuity properties of measurable functions


Chapter 3: Lp-Spaces

Convex functions and inequalities

The Lp-spaces

Approximation by continuous functions


Chapter 4: Elementary Hilbert Space Theory

Inner products and linear functionals

Orthonormal sets

Trigonometric series


Chapter 5: Examples of Banach Space Techniques

Banach spaces

Consequences of Baire's theorem

Fourier series of continuous functions

Fourier coefficients of L1-functions

The Hahn-Banach theorem

An abstract approach to the Poisson integral


Chapter 6: Complex Measures

Total variation

Absolute continuity

Consequences of the Radon-Nikodym theorem

Bounded linear functionals on Lp

The Riesz representation theorem


Chapter 7: Differentiation

Derivatives of measures

The fundamental theorem of Calculus

Differentiable transformations


Chapter 8: Integration on Product Spaces

Measurability on cartesian products

Product measures

The Fubini theorem

Completion of product measures


Distribution functions


Chapter 9: Fourier Transforms

Formal properties

The inversion theorem

The Plancherel theorem

The Banach algebra L1


Chapter 10: Elementary Properties of Holomorphic Functions

Complex differentiation

Integration over paths

The local Cauchy theorem

The power series representation

The open mapping theorem

The global Cauchy theorem

The calculus of residues


Chapter 11: Harmonic Functions

The Cauchy-Riemann equations

The Poisson integral

The mean value property

Boundary behavior of Poisson integrals

Representation theorems


Chapter 12: The Maximum Modulus Principle


The Schwarz lemma

The Phragmen-Lindelöf method

An interpolation theorem

A converse of the maximum modulus theorem


Chapter 13: Approximation by Rational Functions


Runge's theorem

The Mittag-Leffler theorem

Simply connected regions


Chapter 14: Conformal Mapping

Preservation of angles

Linear fractional transformations

Normal families

The Riemann mapping theorem

The class L

Continuity at the boundary

Conformal mapping of an annulus


Chapter 15: Zeros of Holomorphic Functions

Infinite Products

The Weierstrass factorization theorem

An interpolation problem

Jensen's formula

Blaschke products

The Müntz-Szas theorem


Chapter 16: Analytic Continuation

Regular points and singular points

Continuation along curves

The monodromy theorem

Construction of a modular function

The Picard theorem


Chapter 17: Hp-Spaces

Subharmonic functions

The spaces Hp and N

The theorem of F. and M. Riesz

Factorization theorems

The shift operator

Conjugate functions


Chapter 18: Elementary Theory of Banach Algebras


The invertible elements

Ideals and homomorphisms



Chapter 19: Holomorphic Fourier Transforms


Two theorems of Paley and Wiener

Quasi-analytic classes

The Denjoy-Carleman theorem


Chapter 20: Uniform Approximation by Polynomials


Some lemmas

Mergelyan's theorem


Appendix: Hausdorff's Maximality Theorem

Notes and Comments


List of Special Symbols


About the Author

Walter Rudin

Need support?   We're here to help - Get real-world support and resources every step of the way.