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A First Course in Partial Differential Equations with Complex Variables and Transform Methods

H. F. Weinberger
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
William J. Satzer
, on

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 beat-up 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 ε-δ 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 one-dimensional 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 well-posed, and what role do boundary conditions play? The second chapter broadens the discussion to include linear second-order 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 Sturm-Liouville 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 one-lecture-sized, and the topics are well matched to fill out a one-year 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 ( 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.




The one-dimensional wave equation
  1. A physical problem and its mathematical models: the vibrating string
  2. The one-dimensional wave equation
  3. Discussion of the solution: characteristics
  4. Reflection and the free boundary problem
  5. The nonhomogeneous wave equation
II. Linear second-order partial differential equations in two variables
  6. Linearity and superposition
  7. Uniqueness for the vibrating string problem
  8. Classification of second-order equations with constant coefficients
  9. Classification of general second-order 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 Riemann-Lebesgue 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 three-dimensional wave equation in a cube
  35. Poisson's equation in a cube
VII. Sturm-Liouville theory and general Fourier expansions
  36. Eigenfunction expansions for regular second-order 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 three-dimensional 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 one-dimensional 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 initial-boundary value problems
  81. The finite difference method for Laplace's equation
  82. The method of successive approximations
  83. The Raleigh-Ritz method