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## 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)
Contenidos:
``` CONTENTS
Preface v
Chapter I
Suction INTRODUCTION
1. The Two Related Problems 
2. Linear Differential Equations 
3. Infinite Series of Solutions 
4. Boundary Value Problems 
Chapter II
PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS
5. Gravitational Potential 
6. Laplace’s Equation 
7. Cylindrical and Spherical Coordinates 
8. The Flux of Heat 
9. The Heat Equation 
10. Other Cases of the Heat Equation 
11. The Equation of the Vibrating String 
12. Other Equations. Types 
13. A Problem in Vibrations of a String 
14. Example. The Plucked String 
15. The Fourier Sine Series 
16. Imaginary Exponential Functions 
Chapter III
ORTHOGONAL SETS OF FUNCTIONS
17. Inner Product of Two Vectors. Orthogonality 
18. Orthonormal Sets of Vectors 
19. Functions as Vectors. Orthogonality 
20. Generalized Fourier Series 
21. Approximation in the Mean 
22. Closed and Complete Systems 
23. Other Types of Orthogonality 
24. Orthogonal Functions Generated by Differential Equations 
25. Orthogonality of the Characteristic Functions 
Chapter IV FOURIER SERIES
26. Definition 
27. Periodicity of the Function. Example 
28. Fourier Sine Series. Cosine Series 
29. Illustration 
30. Other Forms of Fourier Series 
31. Sectionally Continuous Functions 
32. Preliminary Theory 
33. A Fourier Theorem 
34. Discussion of the Theorem 
35. The Orthonormal Trigonometric Functions 
Chapter V
FURTHER PROPERTIES OF FOURIER SERIES; FOURIER INTEGRALS
36. Differentiation of Fourier Series 
37. Integration of Fourier Series 
38. Uniform Convergence 
39. Concerning More General Conditions 
40. The Fourier Integral 
41. Other Forms of the Fourier Integral 
Chapter VI
SOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS
42. Formal and Rigorous Solutions 
43. The Vibrating String 
44. Variations of the Problem 
45. Temperatures in a Slab with Faces at Temperature Zero 
46. The Above Solution Established. Uniqueness 
47. Variations of the Problem of Temperatures in a Slab 
48. Temperatures in a Sphere 
49. Steady Temperatures in a Rectangular Plate 
50. Displacements in a Membrane. Fourier Series in Two Variables 
51. Temperatures in an Infinite Bar. Application of Fourier Integrals 
52. Temperatures in a Semi-infinite Bar 
53. Further Applications of the Series and Integrals 
Chapter VII
UNIQUENESS OF SOLUTIONS
54. Introduction 
55. Abel’s Test for Uniform Convergence of Series 
56. Uniqueness Theorems for Temperature Problems 
57. Example 
58. Uniqueness of the Potential Function 
59. An Application 
Chapter VIII
BESSEL FUNCTIONS AND APPLICATIONS
60. Derivation of the Functions Jn(x) 
61. The Functions of Integral Orders 
62. Differentiation and Recursion Formulas 
63. Integral Forms of Jn(x) 
64. The Zeros of Jn(x) 
65. The Orthogonality of Bessel Functions 
66. The Orthonormal Functions 
67. Fourier-Bessel Expansions of Functions 
68. Temperatures in an Infinite Cylinder 
69. Radiation at the Surface of the Cylinder 
70. The Vibration of a Circular Membrane 
Chapter IX
LEGENDRE POLYNOMIALS AND APPLICATIONS
71. Derivation of the Legendre Polynomials 
72. Other Legendre Functions 
73. Generating Functions for Pn(x) 
74. The Legendre Coefficients 
75. The Orthogonality of Pn(x). Norms 
76. The Functions Pn(x) as a Complete Orthogonal Set 
77. The Expansion of xm 
78. Derivatives of the Polynomials 
79. Ah Expansion Theorem 
80. The Potential about a Spherical Surface 
81. The Gravitational Potential Due to a Circular Plate 
Index 
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CONTENTS --
Preface v --
Chapter I --
Suction INTRODUCTION --
1. The Two Related Problems  --
2. Linear Differential Equations  --
3. Infinite Series of Solutions  --
4. Boundary Value Problems  --
Chapter II --
PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS --
5. Gravitational Potential  --
6. Laplace’s Equation  --
7. Cylindrical and Spherical Coordinates  --
8. The Flux of Heat  --
9. The Heat Equation  --
10. Other Cases of the Heat Equation  --
11. The Equation of the Vibrating String  --
12. Other Equations. Types  --
13. A Problem in Vibrations of a String  --
14. Example. The Plucked String  --
15. The Fourier Sine Series  --
16. Imaginary Exponential Functions  --
Chapter III --
ORTHOGONAL SETS OF FUNCTIONS --
17. Inner Product of Two Vectors. Orthogonality  --
18. Orthonormal Sets of Vectors  --
19. Functions as Vectors. Orthogonality  --
20. Generalized Fourier Series  --
21. Approximation in the Mean  --
22. Closed and Complete Systems  --
23. Other Types of Orthogonality  --
24. Orthogonal Functions Generated by Differential Equations  --
25. Orthogonality of the Characteristic Functions  --
Chapter IV FOURIER SERIES --
26. Definition  --
27. Periodicity of the Function. Example  --
28. Fourier Sine Series. Cosine Series  --
29. Illustration  --
30. Other Forms of Fourier Series  --
31. Sectionally Continuous Functions  --
32. Preliminary Theory  --
33. A Fourier Theorem  --
34. Discussion of the Theorem  --
35. The Orthonormal Trigonometric Functions  --
Chapter V --
FURTHER PROPERTIES OF FOURIER SERIES; FOURIER INTEGRALS --
36. Differentiation of Fourier Series  --
37. Integration of Fourier Series  --
38. Uniform Convergence  --
39. Concerning More General Conditions  --
40. The Fourier Integral  --
41. Other Forms of the Fourier Integral  --
Chapter VI --
SOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS --
42. Formal and Rigorous Solutions  --
43. The Vibrating String  --
44. Variations of the Problem  --
45. Temperatures in a Slab with Faces at Temperature Zero  --
46. The Above Solution Established. Uniqueness  --
47. Variations of the Problem of Temperatures in a Slab  --
48. Temperatures in a Sphere  --
49. Steady Temperatures in a Rectangular Plate  --
50. Displacements in a Membrane. Fourier Series in Two Variables  --
51. Temperatures in an Infinite Bar. Application of Fourier Integrals  --
52. Temperatures in a Semi-infinite Bar  --
53. Further Applications of the Series and Integrals  --
Chapter VII --
UNIQUENESS OF SOLUTIONS --
54. Introduction  --
55. Abel’s Test for Uniform Convergence of Series  --
56. Uniqueness Theorems for Temperature Problems  --
57. Example  --
58. Uniqueness of the Potential Function  --
59. An Application  --
Chapter VIII --
BESSEL FUNCTIONS AND APPLICATIONS --
60. Derivation of the Functions Jn(x)  --
61. The Functions of Integral Orders  --
62. Differentiation and Recursion Formulas  --
63. Integral Forms of Jn(x)  --
64. The Zeros of Jn(x)  --
65. The Orthogonality of Bessel Functions  --
66. The Orthonormal Functions  --
67. Fourier-Bessel Expansions of Functions  --
68. Temperatures in an Infinite Cylinder  --
69. Radiation at the Surface of the Cylinder  --
70. The Vibration of a Circular Membrane  --
Chapter IX --
LEGENDRE POLYNOMIALS AND APPLICATIONS --
71. Derivation of the Legendre Polynomials  --
72. Other Legendre Functions  --
73. Generating Functions for Pn(x)  --
74. The Legendre Coefficients  --
75. The Orthogonality of Pn(x). Norms  --
76. The Functions Pn(x) as a Complete Orthogonal Set  --
77. The Expansion of xm  --
78. Derivatives of the Polynomials  --
79. Ah Expansion Theorem  --
80. The Potential about a Spherical Surface  --
81. The Gravitational Potential Due to a Circular Plate  --
Index  --

MR, 2,189d

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