Mathematical methods in engineering and physics: special functions and boundary-value problems
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Ronald
1965
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 273 S. |
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MARC
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| 245 | 1 | 0 | |a Mathematical methods in engineering and physics |b special functions and boundary-value problems |c David E. Johnson ; Johnny R. Johnson |
| 264 | 1 | |a New York |b Ronald |c 1965 | |
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Record in the Search Index
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| adam_text | Contents
1 ORTHOGONAL FUNCTIONS I
1.1 Introduction, 1
1.2 Preliminary Definitions, 4
1.3 Generalized Fourier Series, 6
1.4 Bessel s Inequality, 7
1.5 Parseval s Equation, 7
1.6 An Orthogonalization Process, 9
1.7 An Example, 11
1.8 Generating Functions and Recurrence Relations, 13
1.9 Sturm Liouville Systems, 17
1.10 Orthogonal Functions of Two Variables, 20
1.11 Rodrigues Formula, 22
2 FOURIER SERIES 25
2.1 Fourier Trigonometric Series, 25
2.2 An Example, 27
2.3 Even and Odd Functions, 29
2.4 Expansions of Even and Odd Functions; Half Range Series,
30
2.5 Operations with Fourier Series, 34
2.6 Parseval s Theorem, 38
2.7 Fourier Integral, 39
3 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS 42
3.1 Preliminary Example, 42
3.2 Generalized Series Solution, 45
3.3 Existence of the Series Solution, 46
3.4 A Second Solution, 48
4 LEGENDRE FUNCTIONS 51
4.1 Legendre Polynomials, 51
4.2 Legendre Functions of the Second Kind, 54
v
vi CONTENTS
4.3 Generating Function for Pn(x), 55
4.4 Rodrigues Formula, 57
4.5 Orthogonality of the Pn(x), 58
4.6 Recurrence Relations for Pn(x), 59
4.7 Series Expansions Involving Pn(x), 61
4.8 Associated Legendre Functions, 64
4.9 Recurrence Relations for Pnm(x), 65
4.10 Orthogonality and Generating Function of Pnm(x), 67
4.11 Spherical Harmonics, 70
4.12 Series Expansions Involving Pnm{x), 72
4.13 Another Expression for Qn(x), 74
4.14 Recurrence Relations for Qn{x), 75
4.15 Generating Function for Qn(x), 77
5 THE GAMMA FUNCTION 79
5.1 Integral Definition, 79
5.2 Euler s Constant, 81
5.3 Weierstrass Definition, 83
5.4 Other Forms for the Gamma Function, 84
5.5 Logarithmic Derivative, 86
6 BESSEL FUNCTIONS 88
6.1 Bessel s Differential Equation, 88
6.2 Bessel Function of the Second Kind, 91
6.3 Generating Function for Jn{x), 94
6.4 Recurrence Relations, 97
6.5 Spherical Bessel Functions, 99
6.6 Zeros of Jn(x), 101
6.7 Orthogonality of Jn(x), 102
6.8 Integral Relations, 105
6.9 Some Properties of Yn(x), 109
6.10 An Orthogonality Relation Involving Yn(x), 111
7 BOUNDARY VALUE PROBLEMS 113
7.1 Linear Operators and Boundary Value Problems, 113
7.2 Principle of Superposition, 115
7.3 Infinite Series of Solutions, 116
7.4 Separation of Variables Method, 119
7.5 Summary of the Method, 120
7.6 An Example, 122
7.7 Limitations of the Method, 125
8 PARTIAL DIFFERENTIAL EQUATIONS OF
MATHEMATICAL PHYSICS 129
8.1 Helmholtz Equation, 129
8.2 Wave Equation, 131
CONTENTS vii
8.3 Vibrating String, 131
8.4 Vibrating Membrane, 134
8.5 Diffusion Equation, 136
8.6 Laplace s Equation, 140
9 HERMITE POLYNOMIALS 145
9.1 Definition, 145
9.2 Generating Function, 146
9.3 Recurrence Relations, 147
9.4 Orthogonality, 148
9.5 Expansion of Functions in Terms of Hn(x), 149
9.6 General Solution of Hermite s Equation, 151
9.7 Hermite s Orthogonal Functions, 152
10 LAGUERRE POLYNOMIALS 154
10.1 Definition, 154
10.2 Recurrence Relations and Differential Equation, 155
10.3 Rodrigues Formula, 158
10.4 Orthogonality, 158
10.5 Simple Laguerre, Polynomials Ln(x), 160
10.6 Example from Quantum Mechanics, 163
11 CHEBYSHEV POLYNOMIALS 166
11.1 Definitions, 166
11.2 Recurrence Relations and Differential Equations, 167
11.3 Orthogonality Relations, 169
11.4 Generating Functions, 171
11.5 Rodrigues Formula, 173
11.6 Zeros of Tn(x) and Associated Properties, 174
11.7 Expansions in Series of Chebyshev Polynomials, 175
11.8 An Approximation Example, 179
11.9 Boundary Value Problems, 180
12 MATHIEU FUNCTIONS 183
12.1 Mathieu s Equation, 183
12.2 Properties of Elliptic Cylinder Coordinates, 184
12.3 Solution of Mathieu s Equation, 185
12.4 Nature of the General Solutions, 189
12.5 Orthogonality of the Periodic Solutions, 191
12.6 An Example, 193
13 OTHER SPECIAL FUNCTIONS 196
13.1 Hypergeometric Function, 196
13.2 Jacobi Polynomials, 200
13.3 Rodrigues Formula for Jacobi Polynomials, 201
13.4 Orthogonality of the Jacobi Polynomials, 202
13.5 Bessel Polynomials, 205
13.6 Some Related Polynomials, 206
viii CONTENTS
14 LAPLACE AND FOURIER TRANSFORMS 210
14.1 Introduction, 210
14.2 Laplace Transform, 211
14.3 Solutions of Differential Equations, 214
14.4 Convolution, 216
14.5 Fourier Transform, 219
14.6 Properties of the Fourier Transform, 221
14.7 System Functions, 223
14.8 Filter Theory, 224
15 STURM LIOUVILLE TRANSFORMS 229
15.1 Definition, 229
15.2 Finite Fourier Sine and Cosine Transforms, 230
15.3 Hankel Transform, 234
15.4 Legendre Transform, 236
15.5 Laguerre Transform, 238
15.6 Hermite Transform, 239
15.7 Other Transforms, 239
16 A GENERAL CLASS OF ORTHOGONAL
POLYNOMIALS 242
16.1 A Unifying Concept, 242
16.2 Orthogonality of Gn, 244
16.3 Norm of Gn( 1, a + jS, aft h, k, Cn, x), 245
16.4 Infinite Intervals, 248
16.5 Generating Functions, 249
16.6 Summary, 252
APPENDIX
A PROPERTIES OF INFINITE SERIES 255
A.I Convergent Series, 255
A.2 Uniformly Convergent Series, 256
A.3 Power Series, 258
B CONVERGENCE OF THE FOURIER SERIES 259
B.I Sufficiency for Convergence, 259
C TABLES 262
1 Laplace Transforms, 262
2 Finite Sine Transforms, 264
3 Finite Cosine Transforms, 264
4 Summary of Properties of Polynomial Sets { j n(x)}, 265
5 Generating Functions, 266
BIBLIOGRAPHY 267
INDEX 269
|
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| spelling | Johnson, David Edsel Verfasser aut Mathematical methods in engineering and physics special functions and boundary-value problems David E. Johnson ; Johnny R. Johnson New York Ronald 1965 VIII, 273 S. txt rdacontent n rdamedia nc rdacarrier Mathématiques de l'ingénieur Physique mathématique Mathematische Physik Engineering mathematics Mathematical physics Johnson, Johnny Ray Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010151117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Johnson, David Edsel Johnson, Johnny Ray Mathematical methods in engineering and physics special functions and boundary-value problems Mathématiques de l'ingénieur Physique mathématique Mathematische Physik Engineering mathematics Mathematical physics |
| title | Mathematical methods in engineering and physics special functions and boundary-value problems |
| title_auth | Mathematical methods in engineering and physics special functions and boundary-value problems |
| title_exact_search | Mathematical methods in engineering and physics special functions and boundary-value problems |
| title_full | Mathematical methods in engineering and physics special functions and boundary-value problems David E. Johnson ; Johnny R. Johnson |
| title_fullStr | Mathematical methods in engineering and physics special functions and boundary-value problems David E. Johnson ; Johnny R. Johnson |
| title_full_unstemmed | Mathematical methods in engineering and physics special functions and boundary-value problems David E. Johnson ; Johnny R. Johnson |
| title_short | Mathematical methods in engineering and physics |
| title_sort | mathematical methods in engineering and physics special functions and boundary value problems |
| title_sub | special functions and boundary-value problems |
| topic | Mathématiques de l'ingénieur Physique mathématique Mathematische Physik Engineering mathematics Mathematical physics |
| topic_facet | Mathématiques de l'ingénieur Physique mathématique Mathematische Physik Engineering mathematics Mathematical physics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010151117&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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