Mittag-Leffler functions, related topics and applications:
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2014]
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xiv, 443 Seiten |
| ISBN: | 9783662439296 |
| ISSN: | 1439-7382 |
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MARC
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| 100 | 1 | |a Gorenflo, Rudolf |d 1930-2017 |e Verfasser |0 (DE-588)120951339 |4 aut | |
| 245 | 1 | 0 | |a Mittag-Leffler functions, related topics and applications |c Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2014] | |
| 264 | 4 | |c © 2014 | |
| 300 | |a xiv, 443 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics |x 1439-7382 | |
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| 700 | 1 | |a Kilbas, Anatoly A. |d 1948-2010 |e Verfasser |0 (DE-588)1042268665 |4 aut | |
| 700 | 1 | |a Mainardi, Francesco |d 1942- |e Verfasser |0 (DE-588)1055139370 |4 aut | |
| 700 | 1 | |a Rogosin, Sergei V. |d 1952- |e Verfasser |0 (DE-588)1105054837 |4 aut | |
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Record in the Search Index
| _version_ | 1809770648418910208 |
|---|---|
| adam_text |
CONTENTS
1 INTRODUCTION 1
2 HISTORICAL OVERVIEW OF THE MITTAG-LETHER FUNCTIONS 7
2.1 A FEW BIOGRAPHICAL NOTES ON GOSTA MAGNUS MITTAG-LEFFLER 7
2.2 THE CONTENTS OF THE FIVE PAPERS BY MITTAG-LEFFLER
ON NEW FUNCTIONS 9
2.3 FURTHER HISTORY OF MITTAG-LEFFLER FUNCTIONS 12
3 THE CLASSICAL MITTAG-LEFFLER FUNCTION 17
3.1 DEFINITION AND BASIC PROPERTIES 17
3.2 RELATIONS TO ELEMENTARY AND SPECIAL FUNCTIONS 19
3.3 RECURRENCE AND DIFFERENTIAL RELATIONS 21
3.4 INTEGRAL REPRESENTATIONS AND ASYMPTOTICS 23
3.5 DISTRIBUTION OF ZEROS 30
3.6 FURTHER ANALYTIC PROPERTIES 35
3.7 THE MITTAG-LEFFLER FUNCTION OF A REAL VARIABLE 39
3.7.1 INTEGRAL TRANSFORMS 39
3.7.2 THE COMPLETE MONOTONICITY PROPERTY 46
3.7.3 RELATION TO FRACTIONAL CALCULUS 48
3.8 HISTORICAL AND BIBLIOGRAPHICAL NOTES 51
3.9 EXERCISES 53
4 THE TWO-PARAMETRIC MITTAG-LEFFLER FUNCTION 55
4.1 SERIES REPRESENTATION AND PROPERTIES OF COEFFICIENTS 56
4.2 EXPLICIT FORMULAS: RELATIONS TO ELEMENTARY AND SPECIAL
FUNCTIONS 57
4.3 DIFFERENTIAL AND RECURRENCE RELATIONS 58
4.4 INTEGRAL RELATIONS AND ASYMPTOTICS 60
4.5 THE TWO-PARAMETRIC MITTAG-LEFFLER FUNCTION
AS AN ENTIRE FUNCTION 65
4.6 DISTRIBUTION OF ZEROS 67
IX
HTTP://D-NB.INFO/1051867037
X CONTENTS
4.7 COMPUTATIONS WITH THE TWO-PARAMETRIC MITTAG-LEFFLER FUNCTION . 74
4.8 EXTENSION FOR NEGATIVE VALUES OF THE FIRST PARAMETER 80
4.9 FURTHER ANALYTIC PROPERTIES 83
4.10 THE TWO-PARAMETRIC MITTAG-LEFFLER FUNCTION OF A REAL VARIABLE .
84
4.10.1 INTEGRAL TRANSFORMS OF THE TWO-PARAMETRIC
MITTAG-LEFFLER FUNCTION 84
4.10.2 THE COMPLETE MONOTONICITY PROPERTY 85
4.10.3 RELATIONS TO THE FRACTIONAL CALCULUS 86
4.11 HISTORICAL AND BIBLIOGRAPHICAL NOTES 88
4.12 EXERCISES 91
5 MITTAG-LEFLLER FUNCTIONS WITH THREE PARAMETERS 97
5.1 THE PRABHAKAR (THREE-PARAMETRIC MITTAG-LEFFLER) FUNCTION 97
5.1.1 DEFINITION AND BASIC PROPERTIES 97
5.1.2 INTEGRAL REPRESENTATIONS AND ASYMPTOTICS 100
5.1.3 INTEGRAL TRANSFORMS OF THE PRABHAKAR FUNCTION 101
5.1.4 FRACTIONAL INTEGRALS AND DERIVATIVES OF THE
PRABHAKAR FUNCTION 103
5.1.5 RELATIONS TO THE WRIGHT FUNCTION, //-FUNCTION
AND OTHER SPECIAL FUNCTIONS 105
5.2 GENERALIZED (KILBAS-SAIGO) MITTAG-LEFFLER TYPE FUNCTIONS 106
5.2.1 DEFINITION AND BASIC PROPERTIES 106
5.2.2 THE ORDER AND TYPE OF THE ENTIRE FUNCTION E
A
,
M
J(Z) 107
5.2.3 RECURRENCE RELATIONS FOR E
AJ
*J (Z) 110
5.2.4 CONNECTION OF
N
,
M
./(Z) WITH FUNCTIONS
OF HYPERGEOMETRIC TYPE 112
5.2.5 DIFFERENTIATION PROPERTIES OF .,,/(Z) 114
5.2.6 FRACTIONAL INTEGRATION OF THE GENERALIZED
MITTAG-LEFFLER FUNCTION 119
5.2.7 FRACTIONAL DIFFERENTIATION OF THE GENERALIZED
MITTAG-LEFFLER FUNCTION 121
5.3 HISTORICAL AND BIBLIOGRAPHICAL NOTES 125
5.4 EXERCISES 126
6 MULTI-INDEX MITTAG-LEFFLER FUNCTIONS 129
6.1 THE FOUR-PARAMETRIC MITTAG-LEFFLER FUNCTION:
LUCHKO-KILBAS-KIRYAKOVA'S APPROACH 129
6.1.1 DEFINITION AND SPECIAL CASES 129
6.1.2 BASIC PROPERTIES 130
6.1.3 INTEGRAL REPRESENTATIONS AND ASYMPTOTICS 132
6.1.4 EXTENDED FOUR-PARAMETRIC MITTAG-LEFFLER FUNCTIONS 134
6.1.5 RELATIONS TO THE WRIGHT FUNCTION AND THE H-FUNCTION 135
6.1.6 INTEGRAL TRANSFORMS OF THE FOUR-PARAMETRIC
MITTAG-LEFFLER FUNCTION 136
CONTENTS XI
6.1.7 INTEGRAL TRANSFORMS WITH THE FOUR-PARAMETRIC
MITTAG-LEFFLER FUNCTION IN THE KERNEL 138
6.1.8 RELATIONS TO THE FRACTIONAL CALCULUS 142
6.2 MITTAG-LEFFLER FUNCTIONS WITH 2N PARAMETERS 145
6.2.1 DEFINITION AND BASIC PROPERTIES 145
6.2.2 REPRESENTATIONS IN TERMS OF HYPERGEOMETRIC FUNCTIONS. 148
6.2.3 INTEGRAL REPRESENTATIONS AND ASYMPTOTICS 149
6.2.4 EXTENSION OF THE 2/I-PARAMETRIC MITTAG-LEFFLER
FUNCTION 150
6.2.5 RELATIONS TO THE WRIGHT FUNCTION AND TO THE
H-FUNCTION 152
6.2.6 INTEGRAL TRANSFORMS WITH THE MULTI-PARAMETRIC
MITTAG-LEFFLER FUNCTIONS 153
6.2.7 RELATIONS TO THE FRACTIONAL CALCULUS 157
6.3 HISTORICAL AND BIBLIOGRAPHICAL NOTES 159
6.4 EXERCISES 163
7 APPLICATIONS TO FRACTIONAL ORDER EQUATIONS 165
7.1 FRACTIONAL ORDER INTEGRAL EQUATIONS 165
7.1.1 THE ABEL INTEGRAL EQUATION 165
7.1.2 OTHER INTEGRAL EQUATIONS WHOSE SOLUTIONS
ARE REPRESENTED VIA
GENERALIZED MITTAG-LEFFLER
FUNCTIONS 170
7.2 FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS 171
7.2.1 FRACTIONAL ORDINARY DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS 171
7.2.2 ORDINARY FDES WITH VARIABLE COEFFICIENTS 179
7.2.3 OTHER TYPES OF ORDINARY FRACTIONAL DIFFERENTIAL
EQUATIONS 182
7.3 DIFFERENTIAL EQUATIONS WITH FRACTIONAL PARTIAL DERIVATIVES 183
7.3.1 CAUCHY-TYPE PROBLEMS FOR DIFFERENTIAL
EQUATIONS WITH FRACTIONAL PARTIAL DERIVATIVES 184
7.3.2 THE CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS
WITH FRACTIONAL PARTIAL DERIVATIVES 186
7.4 HISTORICAL AND BIBLIOGRAPHICAL NOTES 187
7.5 EXERCISES 197
8 APPLICATIONS TO DETERMINISTIC MODELS 201
8.1 FRACTIONAL RELAXATION AND OSCILLATIONS 201
8.1.1 SIMPLE FRACTIONAL RELAXATION AND OSCILLATION 202
8.1.2 THE COMPOSITE FRACTIONAL RELAXATION AND OSCILLATIONS * 211
8.2 EXAMPLES OF APPLICATIONS OF THE FRACTIONAL CALCULUS
IN PHYSICAL MODELS 219
8.2.1 LINEAR VISCO-ELASTICITY 219
8.2.2 OTHER DETERMINISTIC FRACTIONAL MODELS 224
XII CONTENTS
8.3 HISTORICAL AND BIBLIOGRAPHICAL NOTES 227
8.3.1 GENERAL NOTES 227
8.3.2 NOTES ON FRACTIONAL DIFFERENTIAL EQUATIONS 228
8.3.3 NOTES ON THE FRACTIONAL CALCULUS IN LINEAR
VISCOELASTICITY 228
8.4 EXERCISES 230
9 APPLICATIONS TO STOCHASTIC MODELS 235
9.1 INTRODUCTION 235
9.2 THE MITTAG-LEFFLER PROCESS ACCORDING TO PILLAI 236
9.3 ELEMENTS OF RENEWAL THEORY AND CONTINUOUS TIME
RANDOM WALK (CTRW) 238
9.3.1 RENEWAL PROCESSES 238
9.3.2 CONTINUOUS TIME RANDOM WALK (CTRW) 239
9.3.3 THE RENEWAL PROCESS AS A SPECIAL CTRW 243
9.4 THE POISSON PROCESS AND ITS FRACTIONAL GENERALIZATION
(THE RENEWAL PROCESS OF MITTAG-LEFFLER TYPE) 245
9.4.1 THE MITTAG-LEFFLER WAITING TIME DENSITY 245
9.4.2 THE POISSON PROCESS 246
9.4.3 THE RENEWAL PROCESS OF MITTAG-LEFFLER TYPE 247
9.4.4 THINNING OF A RENEWAL PROCESS 251
9.5 THE FRACTIONAL DIFFUSION PROCESS 253
9.5.1 RENEWAL PROCESS WITH REWARD 253
9.5.2 LIMIT OF THE MITTAG-LEFFLER RENEWAL PROCESS 253
9.5.3 SUBORDINATION IN THE SPACE-TIME FRACTIONAL
DIFFUSION EQUATION 258
9.5.4 THE RESCALING AND RESPEEDING CONCEPT
REVISITED: UNIVERSALITY OF THE MITTAG-LEFFLER DENSITY 262
9.6 HISTORICAL AND BIBLIOGRAPHICAL NOTES 264
9.7 EXERCISES 266
A THE EULERIAN FUNCTIONS 269
A. 1 THE GAMMA FUNCTION 269
A. 1.1 ANALYTIC CONTINUATION 270
A. 1.2 THE GRAPH OF THE GAMMA FUNCTION ON THE REAL AXIS 271
A. 1.3 THE REFLECTION OR COMPLEMENTARY FORMULA 273
A. 1.4 THE MULTIPLICATION FORMULAS 273
A. 1.5 POCHHAMMER'S SYMBOLS 274
A. 1.6 HANKEL INTEGRAL REPRESENTATIONS 274
A. 1.7 NOTABLE INTEGRALS VIA THE GAMMA FUNCTION 276
A. 1.8 ASYMPTOTIC FORMULAS 277
A. 1.9 INFINITE PRODUCTS 277
A.2 THE BETA FUNCTION 278
A.2.1 EULER'S INTEGRAL REPRESENTATION 278
A.2.2 SYMMETRY 278
CONTENTS *'
A.2.3 TRIGONOMETRIC INTEGRAL REPRESENTATION 278
A.2.4 RELATION TO THE GAMMA FUNCTION 279
A.2.5 OTHER INTEGRAL REPRESENTATIONS 279
A.2.6 NOTABLE INTEGRALS VIA THE BETA FUNCTION 280
A.3 HISTORICAL AND BIBLIOGRAPHICAL NOTES 282
A.4 EXERCISES 282
B THE BASICS OF ENTIRE FUNCTIONS 285
B.L DEFINITION AND SERIES REPRESENTATIONS 285
B.2 GROWTH OF ENTIRE FUNCTIONS: ORDER, TYPE AND INDICATOR FUNCTION.
286
B.3 WEIERSTRASS CANONICAL REPRESENTATION: DISTRIBUTION OF ZEROS 287
B.4 ENTIRE FUNCTIONS OF COMPLETELY REGULAR GROWTH 289
B.5 HISTORICAL AND BIBLIOGRAPHICAL NOTES 290
B.6 EXERCISES 292
C INTEGRAL TRANSFORMS 299
C.L FOURIER TYPE TRANSFORMS 299
C.2 THE LAPLACE TRANSFORM 304
C.3 THE MELLIN TRANSFORM 307
C.4 SIMPLE EXAMPLES AND TABLES OF TRANSFORMS OF BASIC
ELEMENTARY AND SPECIAL FUNCTIONS 311
C.5 HISTORICAL AND BIBLIOGRAPHICAL NOTES 313
C.6 EXERCISES 314
D THE MELLIN-BARNES INTEGRAL 319
D.L DEFINITION: CONTOUR OF INTEGRATION 319
D.2 ASYMPTOTIC METHODS FOR THE MELLIN-BARNES INTEGRAL 322
D.3 HISTORICAL AND BIBLIOGRAPHICAL NOTES 324
D.4 EXERCISES 325
E ELEMENTS OF FRACTIONAL CALCULUS 327
E. 1 INTRODUCTION TO THE RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 327
E.2 THE LIOUVILLE-WEYL FRACTIONAL CALCULUS 330
E.3 THE ABEL-RIEMANN FRACTIONAL CALCULUS 334
E.3.1 THE ABEL-RIEMANN FRACTIONAL INTEGRALS AND
DERIVATIVES 334
E.4 THE CAPUTO FRACTIONAL CALCULUS 336
E.4.1 THE CAPUTO FRACTIONAL DERIVATIVE 336
E.5 THE RIESZ-FELLER FRACTIONAL CALCULUS 338
E.5.1 THE RIESZ FRACTIONAL INTEGRALS AND DERIVATIVES 339
E.5.2 THE FELLER FRACTIONAL INTEGRALS AND DERIVATIVES 341
E.6 THE GRIINWALD-LETNIKOV FRACTIONAL CALCULUS 344
E.6.1 THE GRIINWALD-LETNIKOV APPROXIMATION
IN THE RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 345
XIV CONTENTS
E.6.2 THE GRIINWALD-LETNIKOV APPROXIMATION
IN THE RIESZ-FELLER FRACTIONAL CALCULUS 348
E.7 HISTORICAL AND BIBLIOGRAPHICAL NOTES 348
F HIGHER TRANSCENDENTAL FUNCTIONS 351
F. 1 HYPERGEOMETRIC FUNCTIONS 351
F. 1.1 CLASSICAL GAUSS HYPERGEOMETRIC FUNCTIONS 351
F.L
.2 EULER INTEGRAL REPRESENTATION: MELLIN-BARNES
INTEGRAL REPRESENTATION 353
F. 1.3 BASIC PROPERTIES OF HYPERGEOMETRIC FUNCTIONS 354
F. 1.4 THE HYPERGEOMETRIC DIFFERENTIAL EQUATION 355
F.L.5 KUMMER'S AND TRICOMI'S CONFLUENT
HYPERGEOMETRIC FUNCTIONS 356
F. 1.6 GENERALIZED HYPERGEOMETRIC FUNCTIONS
AND THEIR PROPERTIES 360
F.2 WRIGHT FUNCTIONS 361
F.2.1 THE CLASSICAL WRIGHT FUNCTION 361
F.2.2 MELLIN-BARNES INTEGRAL REPRESENTATION AND
ASYMPTOTICS 362
F.2.3 THE BESSEL-WRIGHT FUNCTION: GENERALIZED
WRIGHT FUNCTIONS AND FOX-WRIGHT FUNCTIONS 363
F.3 MEIJER G-FUNCTIONS 365
F.3.1 DEFINITION VIA INTEGRALS: EXISTENCE 365
F.3.2 BASIC PROPERTIES OF THE MEIJER G-FUNCTIONS 366
F.3.3 SPECIAL CASES 367
F.3.4 RELATIONS TO FRACTIONAL CALCULUS 368
F.3.5 INTEGRAL TRANSFORMS OF G-FUNCTIONS 369
F.4 FOX //-FUNCTIONS 370
F.4.1 DEFINITION VIA INTEGRALS: EXISTENCE 370
F.4.2 SERIES REPRESENTATIONS AND ASYMPTOTICS:
RECURRENCE RELATIONS 374
F.4.3 SPECIAL CASES 377
F.4.4 RELATIONS TO FRACTIONAL CALCULUS 379
F.4.5 INTEGRAL TRANSFORMS OF//-FUNCTIONS 381
F.5 HISTORICAL AND BIBLIOGRAPHICAL NOTES 382
F.6 EXERCISES 385
REFERENCES 389
INDEX
441 |
| any_adam_object | 1 |
| author | Gorenflo, Rudolf 1930-2017 Kilbas, Anatoly A. 1948-2010 Mainardi, Francesco 1942- Rogosin, Sergei V. 1952- |
| author_GND | (DE-588)120951339 (DE-588)1042268665 (DE-588)1055139370 (DE-588)1105054837 |
| author_facet | Gorenflo, Rudolf 1930-2017 Kilbas, Anatoly A. 1948-2010 Mainardi, Francesco 1942- Rogosin, Sergei V. 1952- |
| author_role | aut aut aut aut |
| author_sort | Gorenflo, Rudolf 1930-2017 |
| author_variant | r g rg a a k aa aak f m fm s v r sv svr |
| building | Verbundindex |
| bvnumber | BV042176786 |
| classification_rvk | SK 420 |
| ctrlnum | (OCoLC)881388622 (DE-599)DNB1051867037 |
| dewey-full | 515.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.5 |
| dewey-search | 515.5 |
| dewey-sort | 3515.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042176786 |
| illustrated | Not Illustrated |
| indexdate | 2024-09-10T01:29:45Z |
| institution | BVB |
| isbn | 9783662439296 |
| issn | 1439-7382 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027616106 |
| oclc_num | 881388622 |
| open_access_boolean | |
| owner | DE-188 DE-11 DE-83 |
| owner_facet | DE-188 DE-11 DE-83 |
| physical | xiv, 443 Seiten |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Gorenflo, Rudolf 1930-2017 Verfasser (DE-588)120951339 aut Mittag-Leffler functions, related topics and applications Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin Berlin ; Heidelberg Springer [2014] © 2014 xiv, 443 Seiten txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics 1439-7382 Mittag-Leffler-Funktion (DE-588)1065696272 gnd rswk-swf Mittag-Leffler-Funktion (DE-588)1065696272 s DE-604 Kilbas, Anatoly A. 1948-2010 Verfasser (DE-588)1042268665 aut Mainardi, Francesco 1942- Verfasser (DE-588)1055139370 aut Rogosin, Sergei V. 1952- Verfasser (DE-588)1105054837 aut Erscheint auch als Online-Ausgabe 978-3-662-43930-2 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4683793&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027616106&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gorenflo, Rudolf 1930-2017 Kilbas, Anatoly A. 1948-2010 Mainardi, Francesco 1942- Rogosin, Sergei V. 1952- Mittag-Leffler functions, related topics and applications Mittag-Leffler-Funktion (DE-588)1065696272 gnd |
| subject_GND | (DE-588)1065696272 |
| title | Mittag-Leffler functions, related topics and applications |
| title_auth | Mittag-Leffler functions, related topics and applications |
| title_exact_search | Mittag-Leffler functions, related topics and applications |
| title_full | Mittag-Leffler functions, related topics and applications Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |
| title_fullStr | Mittag-Leffler functions, related topics and applications Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |
| title_full_unstemmed | Mittag-Leffler functions, related topics and applications Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin |
| title_short | Mittag-Leffler functions, related topics and applications |
| title_sort | mittag leffler functions related topics and applications |
| topic | Mittag-Leffler-Funktion (DE-588)1065696272 gnd |
| topic_facet | Mittag-Leffler-Funktion |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4683793&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027616106&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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