《Fourier Series and Boundary Value Problems》求取 ⇩

CHAPTER ⅠINTRODUCTION1

1.The Two Related Problems1

2.Linear Differential Equations2

3.Infinite Series of Solutions5

4.Boundary Value Problems6

CHAPTER ⅡPARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS10

5.Gravitational Potential10

6.Laplace’s Equation12

7.Cylindrical and Spherical Coordinates13

8.The Flux of Heat15

9.The Heat Equation17

10.Other Cases of the Heat Equation19

11.The Equation of the Vibrating String21

12.Other Equations.Types23

13.A Problem in Vibrations of a String24

14.Example.The Plucked String28

15.The Fourier Sine Series29

16.Imaginary Exponential Functions31

CHAPTER ⅢORTHOGONAL SETS OF FUNCTIONS34

17.Inner Product of Two Vectors.Orthogonality34

18.Orthonormal Sets of Vectors35

19.Functions as Vectors.Orthogonality37

20.Generalized Fourier Series39

21.Approximation in the Mean40

22.Closed and Complete Systems42

23.Other Types of Orthogonality44

24.Orthogonal Functions Generated by Differential Equations46

25.Orthogonality of the Characteristic Functions49

CHAPTER ⅣFOURIER SERIES53

26.Definition53

27.Periodicity of the Function.Example55

28.Fourier Sine Series.Cosine Series57

29.Illustration59

30.Other Forms of Fourier Series61

31.Sectionally Continuous Functions64

32.Preliminary Theory67

33.A Fourier Theorem70

34.Diacussion of the Theorem72

35.The Orthonormal Trigonometric Functions74

CHAPTER ⅤFURTHER PROPERTIES OF FOURIER SERIES;FOURIER INTEGRALS78

36.Differentiation of Fourier Series78

37.Integration of Fourier Series80

38.Uniform Convergence82

39.Concerning More General Conditions85

40.The Fourier Integral88

41.Other Forms of the Fourier Integral91

CHAPTER ⅥSOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS94

42.Formal and Rigorous Solutions94

43.The Vibrating String95

44.Variations of the Problem98

45.Temperatures in a Slab with Faces at Temperature Zero102

46.The Above Solution Established.Uniqueness105

47.Variations of the Problem of Temperatures in a Slab108

48.Temperatures in a Sphere112

49.Steady Temperatures in a Rectangular Plate114

50.Displacements in a Membrane.Fourier Series in Two Variables116

51.Temperatures in an Infinite Bar.Application of Fourier Integrals120

52.Temperatures in a Semi-infinite Bar122

53.Further Applications of the Series and Integrals123

CHAPTER ⅦUNIQUENESS OF SOLUTIONS127

54.Introduction127

55.Abel’s Test for Uniform Convergence of Series127

56.Uniqueness Theorems for Temperature Problems130

57.Example133

58.Uniqueaess of the Potential Function134

59.An Application137

CHAPTER ⅧBESSEL FUNCTIONS AND APPLICATIONS143

60.Derivation of the Functions Jn(x)143

61.The Functions of Integral Orders145

62.Differentiation and Recursion Formulas148

63.Integral Forms of Jn(x)149

64.The Zeros of Jn(x)153

65.The Orthogonality of Bessel Functions157

66.The Ortbonormal Functions161

67.Fourier-Bessel Expansions of Functions162

68.Temperatures in an Infinite Cylinder165

69.Radiation at the Surface of the Cylinder168

70.The Vibration of a Circular Membrane170

CHAPTER ⅨLEGENDRE POLYNOMIALS AND APPLICATIONS175

71.Derivation of the Legendre Polynomials175

72.Other Legendre Functions177

73.Generating Functions for Pn(x)179

74.The Legendre Coefficients181

75.The Orthogonality of Pn(x).Norms183

76.The Functions Pn(x) as a Complete Orthogonal Set185

77.The Expansion of xm187

78.Derivatives of the Polynomials189

79.An Expansion Theorem191

80.The Potential about a Spherical Surface193

81.The Gravitational Potential Due to a Circular Plate198

INDEX203