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 


