Asymptotics and Mellin-Barnes integrals:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2001
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
85 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 422 S. graph. Darst. |
| ISBN: | 0521790018 |
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MARC
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| adam_text | ENCYCLOPEDIA OFMATHEMATICS AND ITS APPLICATIONS ASYMPTOTICS AND
MELLIN-BARNES INTEGRALS R. B. PARIS D. KAMINSKI CAMBRIDGE UNIVERSITY
PRESS CONTENTS PREFACE PAGE XIII 1 INTRODUCTION 1 1.1 INTRODUCTION TO
ASYMPTOTICS 1 1.1.1 ORDER RELATIONS 1 1.1.2 ASYMPTOTIC EXPANSIONS 4
1.1.3 OTHER EXPANSIONS 15 1.2 BIOGRAPHIES OF MELLIN AND BARNES 25 2
FUNDAMENTAL RESULTS 29 2.1 THE GAMMA FUNCTION T(Z) 29 2.1.1 THE
ASYMPTOTIC EXPANSION OF T(Z) 29 2.1.2 THE STIRLING COEFFICIENTS 32 2.1.3
BOUNDS FOR F(Z) 33 2.2 EXPANSION OF QUOTIENTS OF GAMMA FUNCTIONS 35
2.2.1 INVERSE FACTORIAL EXPANSIONS 36 2.2.2 A RECURSION FORMULA WHEN A R
= FI R * 1 40 2.2.3 EXAMPLES 44 2.2.4 AN ALGEBRAIC METHOD FOR THE
DETERMINATION OF THE AJ 46 2.2.5 SPECIAL CASES 49 2.3 THE ASYMPTOTIC
EXPANSION OF INTEGRAL FUNCTIONS 55 2.3.1 AN EXAMPLE 58 2.4 CONVERGENCE
OF MELLIN-BARNES INTEGRALS 63 2.5 ORDER ESTIMATES FOR REMAINDER
INTEGRALS 69 2.5.1 AN EXAMPLE 69 2.5.2 LEMMAS 71 CONTENTS PROPERTIES
OF MELLIN TRANSFORMS 79 3.1 BASIC PROPERTIES 79 3.1.1 DEFINITION 79
3.1.2 TRANSLATIONAL AND DIFFERENTIAL PROPERTIES ., 81 3.1.3 THE PARSEVAL
FORMULA 82 3.2 ANALYTIC PROPERTIES 85 3.3 INVERSE MELLIN TRANSFORMS 89
3.3.1 INTEGRALS CONNECTED WITH E~ Z 89 3.3.2 SOME STANDARD INTEGRALS 91
3.3.3 DISCONTINUOUS INTEGRALS 93 3.3.4 GAMMA-FUNCTION INTEGRALS 96 3.3.5
RAMAN UJAN-TYPE INTEGRALS 99 3.3.6 BARNES LEMMAS 103 3.4 MELLIN-BARNES
INTEGRAL REPRESENTATIONS 106 3.4.1 THE CONFLUENT HYPERGEOMETRIC
FUNCTIONS 107 3.4.2 THE GAUSS HYPERGEOMETRIC FUNCTION 110 3.4.3 SOME
SPECIAL FUNCTIONS 112 APPLICATIONS OF MELLIN TRANSFORMS 117 4.1
TRANSFORMATION OF SERIES 117 4.1.1 THE MELLIN TRANSFORM METHOD 117 4.1.2
THE POISSON-JACOBI FORMULA 120 4.2 EXAMPLES 122 4.2.1 AN INFINITE SERIES
122 4.2.2 A SMOOTHED DIRICHLET SERIES 125 4.2.3 A FINITE SUM 128 4.3
NUMBER-THEORETIC EXAMPLES 133 4.3.1 A HARMONIC SUM 133 4.3.2 EULER S
PRODUCT 136 4.3.3 RAMANUJAN S FUNCTION 137 4.3.4 SOME OTHER
NUMBER-THEORETIC SUMS 141 4.4 SOLUTION OF DIFFERENTIAL EQUATIONS 146
4.4.1 POTENTIAL PROBLEMS IN WEDGE-SHAPED REGIONS 146 4.4.2 ORDINARY
DIFFERENTIAL EQUATIONS 149 4.4.3 INVERSE MELLIN TRANSFORM SOLUTIONS 152
4.5 SOLUTION OF INTEGRAL EQUATIONS 156 4.5.1 KERNELS OF THE FORM K(XT) *
157 4.5.2 KERNELS OF THE FORM K(X/T) 160 4.6 SOLUTION OF DIFFERENCE
EQUATIONS 164 4.6.1 SOLUTION BY MELLIN TRANSFORMS 166 4.6.2 THE
HYPERGEOMETRIC DIFFERENCE EQUATION 167 4.6.3 SOLUTION OF THE
INHOMOGENEOUS FIRST-ORDER EQUATION 172 CONTENTS IX 4.7 CONVERGENT
INVERSE FACTORIAL SERIES ; 174 5 ASYMPTOTIC EXPANSIONS 179 5.1 ALGEBRAIC
ASYMPTOTIC EXPANSIONS 179 5.1.1 THE EXPONENTIAL INTEGRAL I(Z) 180 5.1.2
THE PARABOLIC CYLINDER FUNCTION D V (Z) 182 5.1.3 A BESSEL FUNCTION
INTEGRAL 184 5.1.4 THE MITTAG-LEFFLER FUNCTION E A (Z) 186 5.2 REMAINDER
INTEGRALS 190 5.2.1 ERROR BOUNDS 190 5.2.2 NUMERICAL EVALUATION 195 5.3
SADDLE-POINT APPROXIMATION OF INTEGRALS 197 5.3.1 AN INTEGRAL DUE TO
HEADING AND WHIPPLE 198 5.3.2 THE BESSEL FUNCTION J N (NX) 200 5.3.3 A
GAUSS HYPERGEOMETRIC FUNCTION 202 5.4 EXPONENTIAL ASYMPTOTIC EXPANSIONS
205 5.4.1 THE EXPONENTIAL INTEGRAL I(Z) 206 5.4.2 THE BESSEL FUNCTION J
V (Z) 20 8 5.4.3 THE PARABOLIC CYLINDER FUNCTION D V {Z) 210 5.4.4 AN
INFINITE SUM 213 5.5 FAXEN S INTEGRAL 216 5.6 INTEGRALS WITH A CONTOUR
BARRIER 220 5.6.1 AN ILLUSTRATIVE EXAMPLE 221 5.6.2 AN INTEGRAL
INVOLVING A BESSEL FUNCTION 225 5.6.3 EXAMPLE IN §4.2.3 REVISITED 229 6
THE STOKES PHENOMENON AND HYPERASYMPTOTICS 234 6.1 THE STOKES PHENOMENON
234 6.1.1 A QUALITATIVE DESCRIPTION 234 6.1.2 THE MODIFIED BESSEL
FUNCTION K V (Z) , 236 6.2 MELLIN-BARNES THEORY 240 6.2.1
EXPONENTIALLY-IMPROVED EXPANSION 241 6.2.2 ESTIMATES FOR R N , M (Z)
244 6.2.3 THE STOKES MULTIPLIER 246 6.2.4 THE STOKES MULTIPLIER FOR A
HIGH-ORDER DIFFERENTIAL EQUATION 249 6.2.5 NUMERICAL EXAMPLES 256 6.2.6
ASYMPTOTICS OF THE TERMINANT T V {Z) 259 6.3 HYPERASYMPTOTICS 265 6.3.1
MELLIN-BARNES THEORY OF HYPERASYMPTOTICS 266 6.3.2 OPTIMAL TRUNCATION
SCHEMES 271 X CONTENTS 6.3.3 A NUMERICAL EXAMPLE 277 6.4
EXPONENTIALLY-IMPROVED ASYMPTOTICS FOR F (Z) 279 6.4.1 ORIGIN OF THE
EXPONENTIALLY SMALL TERMS 280 6.4.2 THE EXPANSION OF Q (Z) 282 6.4.3 A
NUMERICAL EXAMPLE 286 7 MULTIPLE MELLIN-BARNES INTEGRALS 289 7.1 SOME
DOUBLE INTEGRALS 289 7.2 RESIDUES AND DOUBLE INTEGRALS 295 7.3
LAPLACE-TYPE DOUBLE INTEGRALS 299 7.3.1 MELLIN-BARNES INTEGRAL
REPRESENTATION 303 7.3.2 THE NEWTON DIAGRAM 304 7.4 ASYMPTOTICS OF /(A.)
305 7.4.1 TWO INTERNAL POINTS 306 7.4.2 THREE AND MORE INTERNAL POINTS
311 7.4.3 OTHER DOUBLE INTEGRALS 322 7.5 GEOMETRIC CONTENT 323 7.5.1
REMOTENESS 324 7.5.2 ASYMPTOTIC SCALES 326 7.6 LAPLACE-TYPE INTEGRALS OF
HIGHER DIMENSION 328 7.6.1 REPRESENTATION OF TREBLE INTEGRALS 328 7.6.2
ASYMPTOTICS WITH ONE INTERNAL POINT 329 7.6.3 ASYMPTOTICS WITH TWO
INTERNAL POINTS 332 7.6.4 OTHER CONSIDERATIONS FOR TREBLE INTEGRALS 339
7.7 NUMERICAL EXAMPLES 340 8 APPLICATION TO SOME SPECIAL FUNCTIONS 352
8.1 ASYMPTOTICS OF THE EULER-JACOBI SERIES 353 8.1.1 INTRODUCTION 353
8.1.2 THE SADDLE POINT APPROACH 356 8.1.3 THE ASYMPTOTICS OF S P (A) 361
8.1.4 MELLIN-BARNES INTEGRAL APPROACH 365 8.1.5 THE ASYMPTOTIC EXPANSION
OF S P (A) FOR A *** 0+ 368 8.1.6 THE COEFFICIENTS A T 371 8.1.7 THE
STOKES PHENOMENON FOR P ~ 2 374 8.2 AN ASYMPTOTIC FORMULA FOR % ( + IT)
380 8.2.1 INTRODUCTION 380 8.2.2 AN EXPANSION FOR (S) 382 8.2.3 AN
EXPONENTIALLY-SMOOTHED GRAM-TYPE EXPANSION 384 8.2.4 A
RIEMANN-SIEGEL-TYPE EXPANSION 385 8.3 THE ASYMPTOTICS OF PEARCEY S
INTEGRAL 389 8.3.1 INTRODUCTION 389 CONTENTS * XI 8.3.2 A MELLIN-BARNES
INTEGRAL REPRESENTATION 391 8.3.3 ASYMPTOTICS OF P(X, Y) FOR X - OO
393 8.3.4 ASYMPTOTICS OF P(X, Y) FOR Y - OO 398 8.3.5 ASYMPTOTICS OF
P(X, Y) FOR REAL X, Y 400 8.3.6 GENERALISATIONS OF P(X,Y) 401 APPENDIX
404 REFERENCES 409 INDEX 419
|
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| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:01:56Z |
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| isbn | 0521790018 |
| language | English |
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| physical | XVI, 422 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Cambridge Univ. Press |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Paris, Richard B. 1946- Verfasser (DE-588)143980289 aut Asymptotics and Mellin-Barnes integrals R. B. Paris ; D. Kaminski 1. publ. Cambridge Cambridge Univ. Press 2001 XVI, 422 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 85 Asymptotische analyse gtt Développements asymptotiques Integraalrekening gtt Mellin, Transformation de Speciale functies (wiskunde) gtt Asymptotic expansions Mellin transform Mellin-Transformation (DE-588)4339148-5 gnd rswk-swf Asymptotik (DE-588)4126634-1 gnd rswk-swf Asymptotik (DE-588)4126634-1 s Mellin-Transformation (DE-588)4339148-5 s DE-604 Kaminski, David Verfasser aut Encyclopedia of mathematics and its applications 85 (DE-604)BV000903719 85 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009836213&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Paris, Richard B. 1946- Kaminski, David Asymptotics and Mellin-Barnes integrals Encyclopedia of mathematics and its applications Asymptotische analyse gtt Développements asymptotiques Integraalrekening gtt Mellin, Transformation de Speciale functies (wiskunde) gtt Asymptotic expansions Mellin transform Mellin-Transformation (DE-588)4339148-5 gnd Asymptotik (DE-588)4126634-1 gnd |
| subject_GND | (DE-588)4339148-5 (DE-588)4126634-1 |
| title | Asymptotics and Mellin-Barnes integrals |
| title_auth | Asymptotics and Mellin-Barnes integrals |
| title_exact_search | Asymptotics and Mellin-Barnes integrals |
| title_full | Asymptotics and Mellin-Barnes integrals R. B. Paris ; D. Kaminski |
| title_fullStr | Asymptotics and Mellin-Barnes integrals R. B. Paris ; D. Kaminski |
| title_full_unstemmed | Asymptotics and Mellin-Barnes integrals R. B. Paris ; D. Kaminski |
| title_short | Asymptotics and Mellin-Barnes integrals |
| title_sort | asymptotics and mellin barnes integrals |
| topic | Asymptotische analyse gtt Développements asymptotiques Integraalrekening gtt Mellin, Transformation de Speciale functies (wiskunde) gtt Asymptotic expansions Mellin transform Mellin-Transformation (DE-588)4339148-5 gnd Asymptotik (DE-588)4126634-1 gnd |
| topic_facet | Asymptotische analyse Développements asymptotiques Integraalrekening Mellin, Transformation de Speciale functies (wiskunde) Asymptotic expansions Mellin transform Mellin-Transformation Asymptotik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009836213&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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