Fourier series and boundary value problems / by Ruel V. Churchill.
Editor: New York : McGraw-Hill, 1941Descripción: ix, 206 p. ; 24 cmOtra clasificación: 42-01 (35-01)CONTENTS Preface v Chapter I Suction INTRODUCTION 1. The Two Related Problems [1] 2. Linear Differential Equations [2] 3. Infinite Series of Solutions [5] 4. Boundary Value Problems [6] Chapter II PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS 5. Gravitational Potential [10] 6. Laplace’s Equation [12] 7. Cylindrical and Spherical Coordinates [13] 8. The Flux of Heat [15] 9. The Heat Equation [17] 10. Other Cases of the Heat Equation [19] 11. The Equation of the Vibrating String [21] 12. Other Equations. Types [23] 13. A Problem in Vibrations of a String [24] 14. Example. The Plucked String [28] 15. The Fourier Sine Series [29] 16. Imaginary Exponential Functions [31] Chapter III ORTHOGONAL SETS OF FUNCTIONS 17. Inner Product of Two Vectors. Orthogonality [34] 18. Orthonormal Sets of Vectors [35] 19. Functions as Vectors. Orthogonality [37] 20. Generalized Fourier Series [39] 21. Approximation in the Mean [40] 22. Closed and Complete Systems [42] 23. Other Types of Orthogonality [44] 24. Orthogonal Functions Generated by Differential Equations [46] 25. Orthogonality of the Characteristic Functions [49] Chapter IV FOURIER SERIES 26. Definition [53] 27. Periodicity of the Function. Example [55] 28. Fourier Sine Series. Cosine Series [57] 29. Illustration [59] 30. Other Forms of Fourier Series [61] 31. Sectionally Continuous Functions [64] 32. Preliminary Theory [67] 33. A Fourier Theorem [70] 34. Discussion of the Theorem [72] 35. The Orthonormal Trigonometric Functions [74] Chapter V FURTHER PROPERTIES OF FOURIER SERIES; FOURIER INTEGRALS 36. Differentiation of Fourier Series [78] 37. Integration of Fourier Series [80] 38. Uniform Convergence [82] 39. Concerning More General Conditions [85] 40. The Fourier Integral [88] 41. Other Forms of the Fourier Integral [91] Chapter VI SOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS 42. Formal and Rigorous Solutions [94] 43. The Vibrating String [95] 44. Variations of the Problem [98] 45. Temperatures in a Slab with Faces at Temperature Zero [102] 46. The Above Solution Established. Uniqueness [105] 47. Variations of the Problem of Temperatures in a Slab [108] 48. Temperatures in a Sphere [112] 49. Steady Temperatures in a Rectangular Plate [114] 50. Displacements in a Membrane. Fourier Series in Two Variables [116] 51. Temperatures in an Infinite Bar. Application of Fourier Integrals [120] 52. Temperatures in a Semi-infinite Bar [122] 53. Further Applications of the Series and Integrals [123] Chapter VII UNIQUENESS OF SOLUTIONS 54. Introduction [127] 55. Abel’s Test for Uniform Convergence of Series [127] 56. Uniqueness Theorems for Temperature Problems [130] 57. Example [133] 58. Uniqueness of the Potential Function [134] 59. An Application [137] Chapter VIII BESSEL FUNCTIONS AND APPLICATIONS 60. Derivation of the Functions Jn(x) [143] 61. The Functions of Integral Orders [145] 62. Differentiation and Recursion Formulas [148] 63. Integral Forms of Jn(x) [149] 64. The Zeros of Jn(x) [153] 65. The Orthogonality of Bessel Functions [157] 66. The Orthonormal Functions [161] 67. Fourier-Bessel Expansions of Functions [162] 68. Temperatures in an Infinite Cylinder [165] 69. Radiation at the Surface of the Cylinder [168] 70. The Vibration of a Circular Membrane [170] Chapter IX LEGENDRE POLYNOMIALS AND APPLICATIONS 71. Derivation of the Legendre Polynomials [175] 72. Other Legendre Functions [177] 73. Generating Functions for Pn(x) [179] 74. The Legendre Coefficients [181] 75. The Orthogonality of Pn(x). Norms [183] 76. The Functions Pn(x) as a Complete Orthogonal Set [185] 77. The Expansion of xm [187] 78. Derivatives of the Polynomials [189] 79. Ah Expansion Theorem [191] 80. The Potential about a Spherical Surface [193] 81. The Gravitational Potential Due to a Circular Plate [198] Index [203]
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CONTENTS --
Preface v --
Chapter I --
Suction INTRODUCTION --
1. The Two Related Problems [1] --
2. Linear Differential Equations [2] --
3. Infinite Series of Solutions [5] --
4. Boundary Value Problems [6] --
Chapter II --
PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS --
5. Gravitational Potential [10] --
6. Laplace’s Equation [12] --
7. Cylindrical and Spherical Coordinates [13] --
8. The Flux of Heat [15] --
9. The Heat Equation [17] --
10. Other Cases of the Heat Equation [19] --
11. The Equation of the Vibrating String [21] --
12. Other Equations. Types [23] --
13. A Problem in Vibrations of a String [24] --
14. Example. The Plucked String [28] --
15. The Fourier Sine Series [29] --
16. Imaginary Exponential Functions [31] --
Chapter III --
ORTHOGONAL SETS OF FUNCTIONS --
17. Inner Product of Two Vectors. Orthogonality [34] --
18. Orthonormal Sets of Vectors [35] --
19. Functions as Vectors. Orthogonality [37] --
20. Generalized Fourier Series [39] --
21. Approximation in the Mean [40] --
22. Closed and Complete Systems [42] --
23. Other Types of Orthogonality [44] --
24. Orthogonal Functions Generated by Differential Equations [46] --
25. Orthogonality of the Characteristic Functions [49] --
Chapter IV FOURIER SERIES --
26. Definition [53] --
27. Periodicity of the Function. Example [55] --
28. Fourier Sine Series. Cosine Series [57] --
29. Illustration [59] --
30. Other Forms of Fourier Series [61] --
31. Sectionally Continuous Functions [64] --
32. Preliminary Theory [67] --
33. A Fourier Theorem [70] --
34. Discussion of the Theorem [72] --
35. The Orthonormal Trigonometric Functions [74] --
Chapter V --
FURTHER PROPERTIES OF FOURIER SERIES; FOURIER INTEGRALS --
36. Differentiation of Fourier Series [78] --
37. Integration of Fourier Series [80] --
38. Uniform Convergence [82] --
39. Concerning More General Conditions [85] --
40. The Fourier Integral [88] --
41. Other Forms of the Fourier Integral [91] --
Chapter VI --
SOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS --
42. Formal and Rigorous Solutions [94] --
43. The Vibrating String [95] --
44. Variations of the Problem [98] --
45. Temperatures in a Slab with Faces at Temperature Zero [102] --
46. The Above Solution Established. Uniqueness [105] --
47. Variations of the Problem of Temperatures in a Slab [108] --
48. Temperatures in a Sphere [112] --
49. Steady Temperatures in a Rectangular Plate [114] --
50. Displacements in a Membrane. Fourier Series in Two Variables [116] --
51. Temperatures in an Infinite Bar. Application of Fourier Integrals [120] --
52. Temperatures in a Semi-infinite Bar [122] --
53. Further Applications of the Series and Integrals [123] --
Chapter VII --
UNIQUENESS OF SOLUTIONS --
54. Introduction [127] --
55. Abel’s Test for Uniform Convergence of Series [127] --
56. Uniqueness Theorems for Temperature Problems [130] --
57. Example [133] --
58. Uniqueness of the Potential Function [134] --
59. An Application [137] --
Chapter VIII --
BESSEL FUNCTIONS AND APPLICATIONS --
60. Derivation of the Functions Jn(x) [143] --
61. The Functions of Integral Orders [145] --
62. Differentiation and Recursion Formulas [148] --
63. Integral Forms of Jn(x) [149] --
64. The Zeros of Jn(x) [153] --
65. The Orthogonality of Bessel Functions [157] --
66. The Orthonormal Functions [161] --
67. Fourier-Bessel Expansions of Functions [162] --
68. Temperatures in an Infinite Cylinder [165] --
69. Radiation at the Surface of the Cylinder [168] --
70. The Vibration of a Circular Membrane [170] --
Chapter IX --
LEGENDRE POLYNOMIALS AND APPLICATIONS --
71. Derivation of the Legendre Polynomials [175] --
72. Other Legendre Functions [177] --
73. Generating Functions for Pn(x) [179] --
74. The Legendre Coefficients [181] --
75. The Orthogonality of Pn(x). Norms [183] --
76. The Functions Pn(x) as a Complete Orthogonal Set [185] --
77. The Expansion of xm [187] --
78. Derivatives of the Polynomials [189] --
79. Ah Expansion Theorem [191] --
80. The Potential about a Spherical Surface [193] --
81. The Gravitational Potential Due to a Circular Plate [198] --
Index [203] --
MR, 2,189d
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