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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.
This venerable textbook, first published in 1965, has seen broad classroom use at the advanced undergraduate and beginning graduate levels. The Dover edition from 1995 reproduces the original and its contents are identical to those of my old beatup copy from years ago.
The author notes that his effort was motivated by textbooks of the time that focused on boundary value problems, Fourier series and integral transforms, and used these to treat problems in partial differential equations (PDEs). He felt that this gave students the wrong impression about PDEs — for example, that all of them could be handled by separation of variables or integral transforms. So he decided to reverse the sequence and priorities to concentrate on PDEs with supporting materials from transform methods and complex variables. The result is a very nice introduction to partial differential equations and methods of applied mathematics.
The book is intended for advanced undergraduates in mathematics, physics or engineering. The prerequisite is “a good course in elementary calculus”, including convergence and uniform convergence of sequences and series, the εδ definition of a limit, improper integrals, elementary properties of the solutions of ordinary differential equations, partial differentiation, chain rule, gradient, divergence and the divergence theorem. This sounds like more than just “elementary calculus”, doesn’t it? What’s actually needed is a fairly thorough advanced calculus course.
The author begins with the onedimensional wave equation as a model of the vibrating string. This leads quickly to a lot of natural questions: what does it mean to be a solution, do solutions exist, and, if they do, are they unique, what does it mean for a problem to be wellposed, and what role do boundary conditions play? The second chapter broadens the discussion to include linear secondorder PDEs in two variables, and includes classification of equations with constant coefficients. Elliptic and parabolic equations get special attention in the third chapter, which focuses on Laplace’s equation and the heat equation.
An introduction to Fourier series comes next in the context of the heat equation and the method of separation of variables. (When I saw this first as an undergraduate via Seeley’s An Introduction to Fourier Series and Integrals — still available from Dover, I see — I was completely captivated by seeing Fourier’s original approach. It still seems to be a wonderfully direct and intuitive way to introduce Fourier series.) Following this are brief discussions of inhomogeneous equations and Fourier series in multiple dimensions, and then a fairly extensive treatment of SturmLiouville problems.
The next two chapters take up analytic functions of a complex variable and evaluation of integrals via complex variable methods (i.e., the calculus of residues). Inserted here in the book, this material seems to come out of the blue, unmotivated, and it takes a while to get to applications with conformal mapping and an example of its use with Laplace’s equation. The last three chapters include the Laplace and Fourier transforms and a brief introduction to approximation methods.
This book is organized so that most sections are onelecturesized, and the topics are well matched to fill out a oneyear course. The author suggests that this arrangement is especially suited to instructors who are not specialists in partial differential equations. There are lots of exercises. One thing that stands out today is the exclusive emphasis on applications to physics. An instructor using this book now would probably want to include at least some examples from biology and chemistry as well.
Bill Satzer (wjsatzer@mmm.com) 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.
I.

The onedimensional wave equation  
1.  A physical problem and its mathematical models: the vibrating string  
2.  The onedimensional wave equation  
3.  Discussion of the solution: characteristics  
4.  Reflection and the free boundary problem  
5.  The nonhomogeneous wave equation  
II.  Linear secondorder partial differential equations in two variables  
6.  Linearity and superposition  
7.  Uniqueness for the vibrating string problem  
8.  Classification of secondorder equations with constant coefficients  
9.  Classification of general secondorder operators  
III.  Some properties of elliptic and parabolic equations  
10.  Laplace's equation  
11.  Green's theorem and uniqueness for the Laplace's equation  
12.  The maximum principle  
13.  The heat equation  
IV.  Separation of variables and Fourier series  
14.  The method of separation of variables  
15.  Orthogonality and least square approximation  
16.  Completeness and the Parseval equation  
17.  The RiemannLebesgue lemma  
18.  Convergence of the trigonometric Fourier series  
19.  "Uniform convergence, Schwarz's inequality, and completeness"  
20.  Sine and cosine series  
21.  Change of scale  
22.  The heat equation  
23.  Laplace's equation in a rectangle  
24.  Laplace's equation in a circle  
25.  An extension of the validity of these solutions  
26.  The damped wave equation  
V.  Nonhomogeneous problems  
27.  Initial value problems for ordinary differential equations  
28.  Boundary value problems and Green's function for ordinary differential equations  
29.  Nonhomogeneous problems and the finite Fourier transform  
30.  Green's function  
VI.  Problems in higher dimensions and multiple Fourier series  
31.  Multiple Fourier series  
32.  Laplace's equation in a cube  
33.  Laplace's equation in a cylinder  
34.  The threedimensional wave equation in a cube  
35.  Poisson's equation in a cube  
VII.  SturmLiouville theory and general Fourier expansions  
36.  Eigenfunction expansions for regular secondorder ordinary differential equations  
37.  Vibration of a variable string  
38.  Some properties of eigenvalues and eigenfunctions  
39.  Equations with singular endpoints  
40.  Some properties of Bessel functions  
41.  Vibration of a circular membrane  
42.  Forced vibration of a circular membrane: natural frequencies and resonance  
43.  The Legendre polynomials and associated Legendre functions  
44.  Laplace's equation in the sphere  
45.  Poisson's equation and Green's function for the sphere  
VIII.  Analytic functions of a complex variable  
46.  Complex numbers  
47.  Complex power series and harmonic functions  
48.  Analytic functions  
49.  Contour integrals and Cauchy's theorem  
50.  Composition of analytic functions  
51.  Taylor series of composite functions  
52.  Conformal mapping and Laplace's equation  
53.  The bilinear transformation  
54.  Laplace's equation on unbounded domains  
55.  Some special conformal mappings  
56.  The Cauchy integral representation and Liouville's theorem  
IX.  Evaluation of integrals by complex variable methods  
57.  Singularities of analytic functions  
58.  The calculus of residues  
59.  Laurent series  
60.  Infinite integrals  
61.  Infinite series of residues  
62.  Integrals along branch cuts  
X.  The Fourier transform  
63.  The Fourier transform  
64.  Jordan's lemma  
65.  Schwarz's inequality and the triangle inequality for infinite integrals  
66.  Fourier transforms of square integrable functions: the Parseval equation  
67.  Fourier inversion theorems  
68.  Sine and cosine transforms  
69.  Some operational formulas  
70.  The convolution product  
71.  Multiple Fourier transforms: the heat equation in three dimensions  
72.  The threedimensional wave equation  
73.  The Fourier transform with complex argument  
XI.  The Laplace transform  
74.  The Laplace transform  
75.  Initial value problems for ordinary differential equations  
76.  Initial value problems for the onedimensional heat equation  
77.  A diffraction problem  
78.  The Stokes rule and Duhamel's principle  
XII.  Approximation methods  
79.  "Exact" and approximate solutions"  
80.  The method of finite differences for initialboundary value problems  
81.  The finite difference method for Laplace's equation  
82.  The method of successive approximations  
83.  The RaleighRitz method  
SOLUTIONS TO THE EXERCISES  
INDEX  