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Fourier Series and Boundary Value Problems

James Ward Brown and Ruel V. Churchill
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on

Fourier Series and Boundary Value Problems is a classic textbook that was first published in 1941. This edition, the seventh, is a revision of the 2001 edition. Since the third edition James Ward Brown has been co-author, and he has taken over responsibility for revisions since Churchill’s death. The book provides an introductory treatment of Fourier series and their application to boundary value problems in partial differential equations. In its present form, the book is designed to achieve two objectives. The first is to introduce the concept of orthogonal sets of functions and their use in representing arbitrary functions. The second objective is to present the method of separation of variables in the solution of boundary value problems. A basic example is the use of Fourier series to solve the heat equation.

This book is aimed toward students who have had basic courses in ordinary differential equations and advanced calculus. Although it’s completely appropriate for mathematics students, I expect that the primary audience now consists of physics and engineering students. The first summer I was a graduate student, I was assigned as a grader for a course based on — well, an earlier edition of — this book. The course instructor took me aside and warned me that this would actually require me to compute things. (I gather this had been an issue with previous graduate student graders, what with calculation being then somewhat out of fashion.) At that time I very much enjoyed the assignment and the satisfaction of computing Fourier series, expanding functions in series of Bessel functions and Legendre polynomials and solving Sturm-Liouville systems. I think — based on the papers I graded — that most of the students had a similar experience.

This edition shows evidence of considerable revision. The scope of the text has changed very little, although some of the chapters have been rearranged. Most changes seem to be the result of feedback based on classroom experience. The first two chapters now concentrate on Fourier series and their convergence. The chapter on orthogonal functions was rewritten and repositioned so that it is more self-contained and appears in the text right before it is needed. The large collections of exercises have been broken up and moved around so that exercises are matched more clearly with relevant sections of the text.

This continues to be an attractive textbook that is probably still the best choice for learning this subject. The revisions in this most recent edition only enhance its value.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.




1 Fourier Series

Piecewise Continuous Functions

Fourier Cosine Series


Fourier Sine Series


Fourier Series


Adaptations to Other Intervals

2 Convergence of Fourier Series

One-Sided Derivatives

A Property of Fourier Coefficients

Two Lemmas

A Fourier Theorem

Discussion of the Theorem and Its Corollary

Convergence on Other Intervals

A Lemma

Absolute and Uniform Convergence of Fourier Series

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


Boundary Conditions

A Vibrating String

Vibrations of Bars and Membranes

General Solution of the Wave Equation

Types of Equations and Boundary Equations

4 The Fourier Method

Linear Operators

Principle of Superposition

A Temperature Problem

A Vibrating String Problem

Historical Development

5 Boundary Value Problems

A Slab with Faces at Prescribed Temperatures

Related Problems

A Slab with Internally Generated Heat

Steady Temperatures in a Rectangular Plate

Cylindrical Coordinates

A String with Prescribed Initial Conditions


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

More on Superposition of Solutions

Temperatures in a Semi-Infinite Solid

Temperatures in an Unlimited Medium

7 Orthonormal Sets

Inner Products and Orthonormal Sets


Generalized Fourier Series


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


Orthogonality of Eigenfunctions

Real-Valued Eigenfunctions and Nonnegative Eigenvalues

Methods of Solution

Examples of Eigenfunction Expansions

A Temperature Problem in Rectangular Coordinates

Another Problem

Other Coordinates

A Modification of the Method

Another Modification

A Vertically Hung Elastic Bar

9 Bessel Functions and Applications

Bessel Functions Jn(x)

General Solutions of Bessel's Equation

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

The Orthonormal Functions

Fourier-Bessel Series


Temperatures in a Long Cylinder

Internally Generated Heat

Vibration of a Circular Membrane

10 Legendre Polynomials and Applications

Solutions of Legendre's Equation

Legendre Polynomials

Orthogonality of Legendre Polynomials

Rodrigues' Formula and Norms

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



Some Fourier Series Expansions

Solutions of Some Regular Sturm-Liouville Problems