Formulas and theorems for the special functions of mathematical physics:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1966
|
| Ausgabe: | 3., enl. ed. |
| Schriftenreihe: | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
52 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Einheitssacht.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik <engl.> |
| Beschreibung: | VIII, 508 S. |
Internformat
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| 100 | 1 | |a Magnus, Wilhelm |d 1907-1990 |e Verfasser |0 (DE-588)119325721 |4 aut | |
| 245 | 1 | 0 | |a Formulas and theorems for the special functions of mathematical physics |c Wilhelm Magnus ; Fritz Oberhettinger ; Raj Pal Soni |
| 250 | |a 3., enl. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1966 | |
| 300 | |a VIII, 508 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |v 52 | |
| 500 | |a Einheitssacht.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik <engl.> | ||
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Datensatz im Suchindex
| _version_ | 1804136091915124736 |
|---|---|
| adam_text | Contents
Chapter I. The gamma function and related functions 1
1.1 The gamma function 1
1.2 The function ip(z) 13
1.3 The Riemann zeta function £ (z) 19
1.4 The generalized zeta function C{z, a) 22
1.5 Bernoulli and Euler polynomials 25
1.6 Lerch s transcendent (z, s, a) 32
1.7 Miscellaneous results 35
Literature 36
Chapter II. The hypergeometric function 37
2.1 Definitions and elementary relations 37
2.2 The hypergeometric differential equation 42
2.3 Gauss contiguous relations 46
2.4 Linear and higher order transformations 47
2.5 Integral representations 54
2.6 Asymptotic expansions 56
2.7 The Riemann differential equation 57
2.8 Transformation formulas for Riemann s P function 58
2.9 The generalized hypergeometric series 62
2.10 Miscellaneous results 64
Literature 65
Chapter III. Bessel functions 65
3.1 Solutions of the Bessel and the modified Bessel differential
equation 65
3.2 Bessel functions of integer order 69
3.3 Half odd integer order 72
3.4 The Airy functions and related functions 75
3.5 Differential equations and a power series expansion for the pro¬
duct of two Bessel functions 77
3.6 Integral representations for Bessel, Neumann and Hankel func¬
tions 79
3.7 Integral representations for the modified Bessel functions . . 84
3.8 Integrals involving Bessel functions 86
3.9 Addition theorems 106
3.10 Functions related to Bessel functions 108
3.11 Polynomials related to Bessel functions 120
3.12 Series of arbitrary functions in terms of Bessel functions . . . 123
3.13 A list of series involving Bessel functions 129
Contents VII
3.14 Asymptotic expansions 138
3.15 Zeros 146
3.16 Miscellaneous 148
Literature 151
Chapter IV. Legendre functions 151
4.1 Legendre s differential equation 151
4.2 Relations between Legendre functions 164
4.3 The functions P (x) and Ql (%). (Legendre functions on the cut) 166
4.4 Special values for the parameters 172
4.5 Series involving Legendre functions 178
4.6 Integral representations 184
4.7 Integrals involving Legendre functions 191
4.8 Asymptotic behavior 195
4.9 Associated Legendre functions and surface spherical harmonics 198
4.10 Gegenbauer functions, toroidal functions and conical functions 199
Literature 203
Chapter V. Orthogonal polynomials 204
5.1 Orthogonal systems 204
5.2 Jacobi polynomials 209
5.3 Gegenbauer or ultraspherical polynomials 218
5.4 Legendre Polynomials 227
5.5 Generalized Laguerre polynomials 239
5.6 Hermite polynomials 249
5.7 Chebychev (Tchebichef) polynomials 256
Literature 262
Chapter VI. Kummer s function 262
6.1 Definitions and some elementary results 262
6.2 Recurrence relations 267
6.3 The differential equation 268
6.4 Addition and multiplication theorems 271
6.5 Integral representations 274
6.6 Integral transforms associated with 1F1(a; c; z), U(a, c, z) . . 278
6.7 Special cases and its relation to other functions 283
6.8 Asymptotic expansions 288
6.9 Products of Kummer s functions 293
Literature 295
Chapter VII. Whittaker function 295
7.1 Whittaker s differential equation 295
7.2 Some elementary results 301
7.3 Addition and multiplication theorems 306
7.4 Integral representations 311
7.5 Integral transforms 314
7.6 Asymptotic expansions 317
7.7 Products of Whittaker functions 321
Literature 323
VIII Contents
Chapter VIII. Parabolic cylinder functions and parabolic
functions 323
8.1 Parabolic cylinder functions 323
8.2 Parabolic functions 333
Literature 335
Appendix to Chapter VIII 336
Chapter IX. The incomplete gamma function and special
cases 337
9.1 The incomplete gamma function 337
9.2 Special cases 342
Literature 357
Chapter X. Klliptic integrals, theta functions and elliptic
functions 357
10.1 Elliptic integrals 358
10.2 The theta functions 371
10.3 Definition of the Jacobian elliptic functions by the theta func¬
tions 377
10.4 The Jacobian zeta function 386
10.5 The elliptic functions of Weierstrass 387
10.6 Connections between the parameters and special cases .... 392
Literature 395
Chapter XL Integral transforms 395
Examples for the Fourier cosine transform 396
Examples for the Fourier sine transform 397
Examples for the exponential Fourier transform 397
Examples for the Laplace transform 397
Examples for the Mellin transform 397
Examples for the Hankel transform 397
Examples for the Lebedev, Mehler and generalised Mehler transform 398
Example for the Gauss transform 398
11.1 Several examples of solution of integral equations of the first
kind 465
Literature 467
Appendix to Chapter XI 467
Chapter XII. Transformation of systems of coordinates . . 472
12.1 General transformation and special cases 472
12.2 Examples of separation of variables 485
Literature 492
List of special symbols 493
List of functions 495
Index 500
|
| adam_txt |
Contents
Chapter I. The gamma function and related functions 1
1.1 The gamma function 1
1.2 The function ip(z) 13
1.3 The Riemann zeta function £ (z) 19
1.4 The generalized zeta function C{z, a) 22
1.5 Bernoulli and Euler polynomials 25
1.6 Lerch's transcendent (z, s, a) 32
1.7 Miscellaneous results 35
Literature 36
Chapter II. The hypergeometric function 37
2.1 Definitions and elementary relations 37
2.2 The hypergeometric differential equation 42
2.3 Gauss'contiguous relations 46
2.4 Linear and higher order transformations 47
2.5 Integral representations 54
2.6 Asymptotic expansions 56
2.7 The Riemann differential equation 57
2.8 Transformation formulas for Riemann's P function 58
2.9 The generalized hypergeometric series 62
2.10 Miscellaneous results 64
Literature 65
Chapter III. Bessel functions 65
3.1 Solutions of the Bessel and the modified Bessel differential
equation 65
3.2 Bessel functions of integer order 69
3.3 Half odd integer order 72
3.4 The Airy functions and related functions 75
3.5 Differential equations and a power series expansion for the pro¬
duct of two Bessel functions 77
3.6 Integral representations for Bessel, Neumann and Hankel func¬
tions 79
3.7 Integral representations for the modified Bessel functions . . 84
3.8 Integrals involving Bessel functions 86
3.9 Addition theorems 106
3.10 Functions related to Bessel functions 108 '
3.11 Polynomials related to Bessel functions 120
3.12 Series of arbitrary functions in terms of Bessel functions . . . 123
3.13 A list of series involving Bessel functions 129
Contents VII
3.14 Asymptotic expansions 138
3.15 Zeros 146
3.16 Miscellaneous 148
Literature 151
Chapter IV. Legendre functions 151
4.1 Legendre's differential equation 151
4.2 Relations between Legendre functions 164
4.3 The functions P"(x) and Ql (%). (Legendre functions on the cut) 166
4.4 Special values for the parameters 172
4.5 Series involving Legendre functions 178
4.6 Integral representations 184
4.7 Integrals involving Legendre functions 191
4.8 Asymptotic behavior 195
4.9 Associated Legendre functions and surface spherical harmonics 198
4.10 Gegenbauer functions, toroidal functions and conical functions 199
Literature 203
Chapter V. Orthogonal polynomials 204
5.1 Orthogonal systems 204
5.2 Jacobi polynomials 209
5.3 Gegenbauer or ultraspherical polynomials 218
5.4 Legendre Polynomials 227
5.5 Generalized Laguerre polynomials 239
5.6 Hermite polynomials 249
5.7 Chebychev (Tchebichef) polynomials 256
Literature 262
Chapter VI. Kummer's function 262
6.1 Definitions and some elementary results 262
6.2 Recurrence relations 267
6.3 The differential equation 268
6.4 Addition and multiplication theorems 271
6.5 Integral representations 274
6.6 Integral transforms associated with 1F1(a; c; z), U(a, c, z) . . 278
6.7 Special cases and its relation to other functions 283
6.8 Asymptotic expansions 288
6.9 Products of Kummer's functions 293
Literature 295
Chapter VII. Whittaker function 295
7.1 Whittaker's differential equation 295
7.2 Some elementary results 301
7.3 Addition and multiplication theorems 306
7.4 Integral representations 311
7.5 Integral transforms 314
7.6 Asymptotic expansions 317
7.7 Products of Whittaker functions 321
Literature 323
VIII Contents
Chapter VIII. Parabolic cylinder functions and parabolic
functions 323
8.1 Parabolic cylinder functions 323
8.2 Parabolic functions 333
Literature 335
Appendix to Chapter VIII 336
Chapter IX. The incomplete gamma function and special
cases 337
9.1 The incomplete gamma function 337
9.2 Special cases 342
Literature 357
Chapter X. Klliptic integrals, theta functions and elliptic
functions 357
10.1 Elliptic integrals 358
10.2 The theta functions 371
10.3 Definition of the Jacobian elliptic functions by the theta func¬
tions 377
10.4 The Jacobian zeta function 386
10.5 The elliptic functions of Weierstrass 387
10.6 Connections between the parameters and special cases . 392
Literature 395
Chapter XL Integral transforms 395
Examples for the Fourier cosine transform 396
Examples for the Fourier sine transform 397
Examples for the exponential Fourier transform 397
Examples for the Laplace transform 397
Examples for the Mellin transform 397
Examples for the Hankel transform 397
Examples for the Lebedev, Mehler and generalised Mehler transform 398
Example for the Gauss transform 398
11.1 Several examples of solution of integral equations of the first
kind 465
Literature 467
Appendix to Chapter XI 467
Chapter XII. Transformation of systems of coordinates . . 472
12.1 General transformation and special cases 472
12.2 Examples of separation of variables 485
Literature 492
List of special symbols 493
List of functions 495
Index 500 |
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| series | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| series2 | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| spelling | Magnus, Wilhelm 1907-1990 Verfasser (DE-588)119325721 aut Formulas and theorems for the special functions of mathematical physics Wilhelm Magnus ; Fritz Oberhettinger ; Raj Pal Soni 3., enl. ed. Berlin [u.a.] Springer 1966 VIII, 508 S. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 52 Einheitssacht.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik <engl.> Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content Funktion Mathematik (DE-588)4071510-3 s DE-604 Spezielle Funktion (DE-588)4182213-4 s Mathematische Physik (DE-588)4037952-8 s 1\p DE-604 Oberhettinger, Fritz Verfasser aut Soni, Raj P. Verfasser aut Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 52 (DE-604)BV000000395 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Magnus, Wilhelm 1907-1990 Oberhettinger, Fritz Soni, Raj P. Formulas and theorems for the special functions of mathematical physics Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Spezielle Funktion (DE-588)4182213-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4182213-4 (DE-588)4071510-3 (DE-588)4037952-8 (DE-588)4155008-0 |
| title | Formulas and theorems for the special functions of mathematical physics |
| title_auth | Formulas and theorems for the special functions of mathematical physics |
| title_exact_search | Formulas and theorems for the special functions of mathematical physics |
| title_exact_search_txtP | Formulas and theorems for the special functions of mathematical physics |
| title_full | Formulas and theorems for the special functions of mathematical physics Wilhelm Magnus ; Fritz Oberhettinger ; Raj Pal Soni |
| title_fullStr | Formulas and theorems for the special functions of mathematical physics Wilhelm Magnus ; Fritz Oberhettinger ; Raj Pal Soni |
| title_full_unstemmed | Formulas and theorems for the special functions of mathematical physics Wilhelm Magnus ; Fritz Oberhettinger ; Raj Pal Soni |
| title_short | Formulas and theorems for the special functions of mathematical physics |
| title_sort | formulas and theorems for the special functions of mathematical physics |
| topic | Spezielle Funktion (DE-588)4182213-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Spezielle Funktion Funktion Mathematik Mathematische Physik Formelsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
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