Special functions:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
1999
|
| Ausgabe: | 1. publication |
| Schriftenreihe: | Encyclopedia of Mathematics and Its Applications
71 |
| Schlagworte: | |
| Online-Zugang: | Table of contents Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XVI, 664 Seiten graph. Darst. |
| ISBN: | 0521623219 0521789885 9780521623216 9780521789882 |
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| 650 | 4 | |a Hipergeometrik fonksiyonlar | |
| 650 | 4 | |a Hipergeometrik seriler | |
| 650 | 4 | |a Spezielle Funktion | |
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Datensatz im Suchindex
| _version_ | 1804127217249157120 |
|---|---|
| adam_text | Titel: Special functions
Autor: Andrews, George E
Jahr: 2000
Contents
Preface page xiii
1 The Gamma and Beta Functions 1
1.1 The Gamma and Beta Integrals and Functions 2
1.2 The Euler Reflection Formula 9
1.3 The Hurwitz and Riemann Zeta Functions 15
1.4 Stirling s Asymptotic Formula 18
1.5 Gauss s Multiplication Formula for T(mx) 22
1.6 Integral Representations for Log T(x) and (x) 26
1.7 Rummer s Fourier Expansion of Log T (x) 29
1.8 Integrals of Dirichlet and Volumes of Ellipsoids 32
1.9 The Bohr-Mollerup Theorem 34
1.10 Gauss and Jacobi Sums 36
1.11 A Probabilistic Evaluation of the Beta Function 43
1.12 The p-adic Gamma Function 44
Exercises 46
2 The Hypergeometric Functions 61
2.1 The Hypergeometric Series 61
2.2 Euler s Integral Representation 65
2.3 The Hypergeometric Equation 73
2.4 The Barnes Integral for the Hypergeometric Function 85
2.5 Contiguous Relations 94
2.6 Dilogarithms 102
2.7 Binomial Sums 107
2.8 Dougall s Bilateral Sum 109
2.9 Fractional Integration by Parts and Hypergeometric Integrals 111
Exercises 114
3 Hypergeometric Transformations and Identities 124
3.1 Quadratic Transformations 125
3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 132
3.3 Transformations of Balanced Series 140
3.4 Whipple s Transformation 143
3.5 Dougall s Formula and Hypergeometric Identities 147
3.6 Integral Analogs of Hypergeometric Sums 150
3.7 Contiguous Relations 154
3.8 The Wilson Polynomials 157
3.9 Quadratic Transformations - Riemann s View 160
3.10 Indefinite Hypergeometric Summation 163
3.11 The W-Z Method 166
3.12 Contiguous Relations and Summation Methods 174
Exercises 176
4 Hessel Functions and Confluent Hypergeometric Functions 187
4.1 The Confluent Hypergeometric Equation 188
4.2 Barnes s Integral for 1 Fi 192
4.3 Whittaker Functions 195
4.4 Examples of i F] and Whittaker Functions 196
4.5 Bessel s Equation and Bessel Functions 199
4.6 Recurrence Relations 202
4.7 Integral Representations of Bessel Functions 203
4.8 Asymptotic Expansions 209
4.9 Fourier Transforms and Bessel Functions 210
4.10 Addition Theorems 213
4.11 Integrals of Bessel Functions 216
4.12 The Modified Bessel Functions 222
4.13 Nicholson s Integral 223
4.14 Zeros of Bessel Functions 225
4.15 Monotonicity Properties of Bessel Functions 229
4.16 Zero-Free Regions for , F Functions 231
Exercises 234
5 Orthogonal Polynomials 240
5.1 Chebyshev Polynomials 240
5.2 Recurrence 244
5.3 Gauss Quadrature 248
5.4 Zeros of Orthogonal Polynomials 253
5.5 Continued Fractions 256
5.6 Kernel Polynomials 259
5.7 Parseval s Formula 263
5.8 The Moment-Generating Function 266
Exercises 269
6 Special Orthogonal Polynomials 277
6.1 Hermite Polynomials 278
6.2 Laguerre Polynomials 282
6.3 Jacobi Polynomials and Gram Determinants 293
6.4 Generating Functions for Jacobi Polynomials 297
6.5 Completeness of Orthogonal Polynomials 306
6.6 Asymptotic Behavior of P{na,ti){x) for Large n 310
6.7 Integral Representations of Jacobi Polynomials 313
6.8 Linearization of Products of Orthogonal Polynomials 316
6.9 Matching Polynomials 323
6.10 The Hypergeometric Orthogonal Polynomials 330
6.11 An Extension of the Ultraspherical Polynomials 334
Exercises 339
7 Topics in Orthogonal Polynomials 355
7.1 Connection Coefficients 356
7.2 Rational Functions with Positive Power Series Coefficients 363
7.3 Positive Polynomial Sums from Quadrature
and Vietoris s Inequality 371
7.4 Positive Polynomial Sums and the Bieberback Conjecture 381
7.5 A Theorem of Turân 384
7.6 Positive Summability of Ultraspherical Polynomials 388
7.7 The Irrationality of Ç(3) 391
Exercises 395
8 The Selberg Integral and Its Applications 401
8.1 Selberg s and Aomoto s Integrals 402
8.2 Aomoto s Proof of Selberg s Formula 402
8.3 Extensions of Aomoto s Integral Formula 407
8.4 Anderson s Proof of Selberg s Formula 411
8.5 A Problem of Stieltjes and the Discriminant of
a Jacobi Polynomial 415
8.6 Siegel s Inequality 419
8.7 The Stieltjes Problem on the Unit Circle 425
8.8 Constant-Term Identities 426
8.9 Nearly Poised 362 Identities 428
8.10 The Hasse-Davenport Relation 430
8.11 A Finite-Field Analog of Selberg s Integral 434
Exercises 439
9 Spherical Harmonics 445
9.1 Harmonic Polynomials 445
9.2 The Laplace Equation in Three Dimensions 447
9.3 Dimension of the Space of Harmonic Polynomials
of Degree k 449
9.4 Orthogonality of Harmonic Polynomials 451
9.5 Action of an Orthogonal Matrix 452
9.6 The Addition Theorem 454
9.7 The Funk-Hecke Formula 458
9.8 The Addition Theorem for Ultraspherical Polynomials 459
9.9 The Poisson Kernel and Dirichlet Problem 463
9.10 Fourier Transforms 464
9.11 Finite-Dimensional Representations of Compact Groups 466
9.12 The Group SU (2) 469
9.13 Representations of SU (2) 471
9.14 Jacobi Polynomials as Matrix Entries 473
9.15 An Addition Theorem 474
9.16 Relation of SU(2) to the Rotation Group SO(3) 476
Exercises 478
10 Introduction to ^-Series 481
10.1 The «/-Integral 485
10.2 The ^-Binomial Theorem 487
10.3 The g-Gamma Function 493
10.4 The Triple Product Identity 496
10.5 Ramanujan s Summation Formula 501
10.6 Representations of Numbers as Sums of Squares 506
10.7 Elliptic and Theta Functions 508
10.8 q -Beta Integrals 513
10.9 Basic Hypergeometric Series 520
10.10 Basic Hypergeometric Identities 523
10.11 #-Ultraspherical Polynomials 527
10.12 Mellin Transforms 532
Exercises 542
11 Partitions 553
11.1 Background on Partitions 553
11.2 Partition Analysis 555
11.3 A Library for the Partition Analysis Algorithm 557
11.4 Generating Functions 559
11.5 Some Results on Partitions 563
11.6 Graphical Methods 565
11.7 Congruence Properties of Partitions 569
Exercises 573
12 Bailey Chains 577
12.1 Rogers s Second Proof of the Rogers-Ramanujan Identities 577
12.2 Bailey s Lemma 582
12.3 Watson s Transformation Formula 586
12.4 Other Applications 589
Exercises 590
A Infinite Products 595
A.l Infinite Products 595
Exercises 597
B Summability and Fractional Integration 599
B.l Abel and Ce aro Means 599
B.2 The Ce aro Means (C, a) 602
B.3 Fractional Integrals 604
B.4 Historical Remarks 605
Exercises 607
C Asymptotic Expansions 611
C.l Asymptotic Expansion 611
C.2 Properties of Asymptotic Expansions 612
C.3 Watson s Lemma 614
C.4 The Ratio of Two Gamma Functions 615
Exercises 616
D Euler-Maclaurin Summation Formula 617
D.l Introduction 617
D.2 The Euler-Maclaurin Formula 619
D.3 Applications 621
D.4 The Poisson Summation Formula 623
Exercises 627
E Lagrange Inversion Formula 629
E.l Reversion of Series 629
E.2 A Basic Lemma 630
E.3 Lambert s Identity 631
E.4 Whipple s Transformation 632
Exercises 634
F Series Solutions of Differential Equations 637
F.l Ordinary Points 637
F.2 Singular Points 638
F.3 Regular Singular Points 639
Bibliography 641
Index 655
Subject Index 659
Symbol Index 661
|
| any_adam_object | 1 |
| author | Andrews, George E. 1938- Askey, Richard 1933-2019 Roy, Ranjan 1948-2020 |
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| author_facet | Andrews, George E. 1938- Askey, Richard 1933-2019 Roy, Ranjan 1948-2020 |
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| author_sort | Andrews, George E. 1938- |
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| discipline | Mathematik |
| edition | 1. publication |
| format | Book |
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| indexdate | 2024-07-09T18:29:50Z |
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| isbn | 0521623219 0521789885 9780521623216 9780521789882 |
| language | English |
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| series | Encyclopedia of Mathematics and Its Applications |
| series2 | Encyclopedia of Mathematics and Its Applications |
| spelling | Andrews, George E. 1938- Verfasser (DE-588)122581342 aut Special functions George E. Andrews, Richard Askey, Ranjan Roy 1. publication Cambridge [u.a.] Cambridge University Press 1999 XVI, 664 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of Mathematics and Its Applications 71 Hier auch später erschienene, unveränderte Nachdrucke Fonksiyonlar, Özel Hipergeometrik fonksiyonlar Hipergeometrik seriler Spezielle Funktion Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Gammafunktion (DE-588)4289118-8 gnd rswk-swf Orthogonale Polynome (DE-588)4172863-4 gnd rswk-swf Hypergeometrische Reihe (DE-588)4161061-1 gnd rswk-swf Bessel-Funktionen (DE-588)4069359-4 gnd rswk-swf Spezielle Funktion (DE-588)4182213-4 s DE-604 Hypergeometrische Reihe (DE-588)4161061-1 s Bessel-Funktionen (DE-588)4069359-4 s Gammafunktion (DE-588)4289118-8 s Orthogonale Polynome (DE-588)4172863-4 s Askey, Richard 1933-2019 Verfasser (DE-588)129627631 aut Roy, Ranjan 1948-2020 Verfasser (DE-588)1022226789 aut Encyclopedia of Mathematics and Its Applications 71 (DE-604)BV000903719 71 http://www.loc.gov/catdir/toc/cam024/98025757.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008535287&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Andrews, George E. 1938- Askey, Richard 1933-2019 Roy, Ranjan 1948-2020 Special functions Encyclopedia of Mathematics and Its Applications Fonksiyonlar, Özel Hipergeometrik fonksiyonlar Hipergeometrik seriler Spezielle Funktion Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd Gammafunktion (DE-588)4289118-8 gnd Orthogonale Polynome (DE-588)4172863-4 gnd Hypergeometrische Reihe (DE-588)4161061-1 gnd Bessel-Funktionen (DE-588)4069359-4 gnd |
| subject_GND | (DE-588)4182213-4 (DE-588)4289118-8 (DE-588)4172863-4 (DE-588)4161061-1 (DE-588)4069359-4 |
| title | Special functions |
| title_auth | Special functions |
| title_exact_search | Special functions |
| title_full | Special functions George E. Andrews, Richard Askey, Ranjan Roy |
| title_fullStr | Special functions George E. Andrews, Richard Askey, Ranjan Roy |
| title_full_unstemmed | Special functions George E. Andrews, Richard Askey, Ranjan Roy |
| title_short | Special functions |
| title_sort | special functions |
| topic | Fonksiyonlar, Özel Hipergeometrik fonksiyonlar Hipergeometrik seriler Spezielle Funktion Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd Gammafunktion (DE-588)4289118-8 gnd Orthogonale Polynome (DE-588)4172863-4 gnd Hypergeometrische Reihe (DE-588)4161061-1 gnd Bessel-Funktionen (DE-588)4069359-4 gnd |
| topic_facet | Fonksiyonlar, Özel Hipergeometrik fonksiyonlar Hipergeometrik seriler Spezielle Funktion Functions, Special Gammafunktion Orthogonale Polynome Hypergeometrische Reihe Bessel-Funktionen |
| url | http://www.loc.gov/catdir/toc/cam024/98025757.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008535287&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT andrewsgeorgee specialfunctions AT askeyrichard specialfunctions AT royranjan specialfunctions |