Hadamard Expansions and Hyperasymptotic Evaluation :: an Extension of the Method of Steepest Descents /
"The author describes the recently developed theory of Hadamard expansions applied to the high-precision (hyperasymptotic) evaluation of Laplace and Laplace-type integrals. This brand new method builds on the well-known asymptotic method of steepest descents, of which the opening chapter gives...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge ; New York :
Cambridge University Press,
2011, ©2011.
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications ;
v. 141. |
| Schlagworte: | |
| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | "The author describes the recently developed theory of Hadamard expansions applied to the high-precision (hyperasymptotic) evaluation of Laplace and Laplace-type integrals. This brand new method builds on the well-known asymptotic method of steepest descents, of which the opening chapter gives a detailed account illustrated by a series of examples of increasing complexity. A discussion of uniformity problems associated with various coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows. The remaining chapters deal with the Hadamard expansion of Laplace integrals, with and without saddle points. Problems of different types of saddle coalescence are also discussed. The text is illustrated with many numerical examples, which help the reader to understand the level of accuracy achievable. The author also considers applications to some important special functions. This book is ideal for graduate students and researchers working in asymptotics"-- |
| Beschreibung: | 1 online resource (viii, 243 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 235-240) and index. |
| ISBN: | 9781107089853 1107089859 9780511753626 0511753624 9781107096134 1107096138 9781107101722 1107101727 1139887246 9781139887243 1107104165 9781107104167 1107093066 9781107093065 |
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| 100 | 1 | |a Paris, R. B. |q (Richard Bruce), |d 1946- | |
| 245 | 1 | 0 | |a Hadamard Expansions and Hyperasymptotic Evaluation : |b an Extension of the Method of Steepest Descents / |c R.B. Paris. |
| 260 | |a Cambridge ; |a New York : |b Cambridge University Press, |c 2011, ©2011. | ||
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| 490 | 1 | |a Encyclopedia of Mathematics And Its Applications ; |v 141 | |
| 520 | |a "The author describes the recently developed theory of Hadamard expansions applied to the high-precision (hyperasymptotic) evaluation of Laplace and Laplace-type integrals. This brand new method builds on the well-known asymptotic method of steepest descents, of which the opening chapter gives a detailed account illustrated by a series of examples of increasing complexity. A discussion of uniformity problems associated with various coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows. The remaining chapters deal with the Hadamard expansion of Laplace integrals, with and without saddle points. Problems of different types of saddle coalescence are also discussed. The text is illustrated with many numerical examples, which help the reader to understand the level of accuracy achievable. The author also considers applications to some important special functions. This book is ideal for graduate students and researchers working in asymptotics"-- |c Provided by publisher | ||
| 504 | |a Includes bibliographical references (pages 235-240) and index. | ||
| 505 | 0 | |a Preface; 1. Asymptotics of Laplace-type integrals; 2. Hadamard expansion of Laplace integrals; 3. Hadamard expansion of Laplace-type integrals; 4. Applications. | |
| 588 | 0 | |a Print version record. | |
| 546 | |a English. | ||
| 650 | 0 | |a Integral equations |x Asymptotic theory. |0 http://id.loc.gov/authorities/subjects/sh85067089 | |
| 650 | 0 | |a Asymptotic expansions. |0 http://id.loc.gov/authorities/subjects/sh85009056 | |
| 650 | 4 | |a Integral equations |x Asymptotic theory. | |
| 650 | 4 | |a Asymptotic expansions. | |
| 650 | 6 | |a Équations intégrales |x Théorie asymptotique. | |
| 650 | 6 | |a Développements asymptotiques. | |
| 650 | 7 | |a MATHEMATICS |x Algebra |x Abstract. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
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| 650 | 7 | |a Asymptotic expansions |2 fast | |
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| 758 | |i has work: |a Hadamard expansions and hyperasymptotic evaluation (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGMWh9J9FfMqBgC7qFdTXm |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Paris, R.B. (Richard Bruce), 1946- |t Hadamard Expansions and Hyperasymptotic Evaluation. |d Cambridge ; New York : Cambridge University Press, 2011, ©2011 |z 9781107002586 |w (DLC) 2010051563 |w (OCoLC)694393863 |
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| 880 | 0 | |6 505-00/(S |a Cover -- Half Title -- Series Page -- Title -- Copyright -- Contents -- Preface -- 1 Asymptotics of Laplace-type integrals -- 1.1 Historical introduction -- 1.2 The method of steepest descents -- 1.2.1 Preliminaries -- 1.2.2 Asymptotic expansion of I (λ) -- 1.2.3 Quadratic and linear endpoint cases -- 1.2.4 Watson's lemma -- 1.2.5 Approximate methods -- The saddle-point method -- The method of stationary phase -- 1.3 Examples -- 1.4 Further examples -- 1.5 Uniform expansions -- 1.5.1 Saddle point near a pole -- 1.5.2 Saddle point near an endpoint -- 1.5.3 Two coalescing saddle points -- 1.6 Optimal truncation and superasymptotics -- 1.6.1 Optimal truncation -- 1.6.2 Ursell's lemma -- 1.7 The Stokes phenomenon -- 1.7.1 Description of the Stokes phenomenon -- 1.7.2 A rigorous approach -- 1.7.3 The Stokes phenomenon and steepest descents -- 1.8 Hyperasymptotics -- 1.8.1 Overview -- 1.8.2 Airey's converging factors -- 1.8.3 Dingle's converging factors -- 1.8.4 A formal discussion of hyperasymptotics -- 1.8.5 Truncation schemes -- 1.8.6 Hyperasymptotics of Laplace-type integrals -- 2 Hadamard expansion of Laplace integrals -- 2.1 Introduction -- 2.2 The Hadamard series for Iν(x) -- 2.2.1 Derivation of the single-stage expansion -- 2.2.2 The modified Hadamard series -- 2.2.3 The Hadamard series for complex z -- 2.2.4 Multi-stage expansions for Iν(z) -- 2.2.5 The Stokes phenomenon -- 2.2.6 The Hadamard series for 1F1(a -- a + b -- z) -- 2.3 Rapidly convergent Hadamard series -- 2.4 Hadamard series on an infinite interval -- 2.4.1 Subdivision of the integration path -- 2.4.2 The Hadamard expansion -- 2.4.3 Choice of the Ωn -- 2.4.4 A numerical example -- 2.4.5 A computational simplification -- 2.5 Examples -- 2.6 Bounds on the tails of Hadamard series -- 3 Hadamard expansion of Laplace-type integrals -- 3.1 Introduction -- 3.2 Expansion schemes. | |
| 880 | 8 | |6 505-00/(S |a 3.2.1 Subdivision of the u-axis -- 3.2.2 Expansion Scheme A -- 3.2.3 Expansion Scheme B -- 3.3 Examples -- Hadamard expansion near the Stokes line -- 3.4 Coalescence problems -- 3.4.1 Expansion scheme for coalescence problems -- 3.5 Examples of coalescence -- 4 Applications -- 4.1 Introduction -- 4.2 The Bessel function Jν(νz) -- 4.2.1 The case z positive -- 4.2.2 Numerical results -- 4.2.3 The case x -̃ 1 -- 4.2.4 The case z complex -- 4.3 The Pearcey integral -- 4.3.1 The Hadamard expansion of Pe(x, y) -- 4.3.2 Computation in the neighbourhood of the cusps -- 4.4 The parabolic cylinder function -- 4.5 The expansion for log Г(z) -- Appendix A: Properties of P(a, z) -- Appendix B: Convergence of Hadamard series -- Appendix C: Connection with the exp-arc integrals -- References -- Index. | |
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| author | Paris, R. B. (Richard Bruce), 1946- |
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| contents | Preface; 1. Asymptotics of Laplace-type integrals; 2. Hadamard expansion of Laplace integrals; 3. Hadamard expansion of Laplace-type integrals; 4. Applications. |
| ctrlnum | (OCoLC)847526828 |
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(Richard Bruce), 1946-</subfield><subfield code="t">Hadamard Expansions and Hyperasymptotic Evaluation.</subfield><subfield code="d">Cambridge ; New York : Cambridge University Press, 2011, ©2011</subfield><subfield code="z">9781107002586</subfield><subfield code="w">(DLC) 2010051563</subfield><subfield code="w">(OCoLC)694393863</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Encyclopedia of mathematics and its applications ;</subfield><subfield code="v">v. 141.</subfield><subfield code="0">http://id.loc.gov/authorities/names/n42010632</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-862</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=569364</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-863</subfield><subfield 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| id | ZDB-4-EBA-ocn847526828 |
| illustrated | Illustrated |
| indexdate | 2025-03-18T14:21:14Z |
| institution | BVB |
| isbn | 9781107089853 1107089859 9780511753626 0511753624 9781107096134 1107096138 9781107101722 1107101727 1139887246 9781139887243 1107104165 9781107104167 1107093066 9781107093065 |
| language | English |
| oclc_num | 847526828 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (viii, 243 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications ; |
| series2 | Encyclopedia of Mathematics And Its Applications ; |
| spelling | Paris, R. B. (Richard Bruce), 1946- Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / R.B. Paris. Cambridge ; New York : Cambridge University Press, 2011, ©2011. 1 online resource (viii, 243 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier data file Encyclopedia of Mathematics And Its Applications ; 141 "The author describes the recently developed theory of Hadamard expansions applied to the high-precision (hyperasymptotic) evaluation of Laplace and Laplace-type integrals. This brand new method builds on the well-known asymptotic method of steepest descents, of which the opening chapter gives a detailed account illustrated by a series of examples of increasing complexity. A discussion of uniformity problems associated with various coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows. The remaining chapters deal with the Hadamard expansion of Laplace integrals, with and without saddle points. Problems of different types of saddle coalescence are also discussed. The text is illustrated with many numerical examples, which help the reader to understand the level of accuracy achievable. The author also considers applications to some important special functions. This book is ideal for graduate students and researchers working in asymptotics"-- Provided by publisher Includes bibliographical references (pages 235-240) and index. Preface; 1. Asymptotics of Laplace-type integrals; 2. Hadamard expansion of Laplace integrals; 3. Hadamard expansion of Laplace-type integrals; 4. Applications. Print version record. English. Integral equations Asymptotic theory. http://id.loc.gov/authorities/subjects/sh85067089 Asymptotic expansions. http://id.loc.gov/authorities/subjects/sh85009056 Integral equations Asymptotic theory. Asymptotic expansions. Équations intégrales Théorie asymptotique. Développements asymptotiques. MATHEMATICS Algebra Abstract. bisacsh MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Asymptotic expansions fast Integral equations Asymptotic theory fast has work: Hadamard expansions and hyperasymptotic evaluation (Text) https://id.oclc.org/worldcat/entity/E39PCGMWh9J9FfMqBgC7qFdTXm https://id.oclc.org/worldcat/ontology/hasWork Print version: Paris, R.B. (Richard Bruce), 1946- Hadamard Expansions and Hyperasymptotic Evaluation. Cambridge ; New York : Cambridge University Press, 2011, ©2011 9781107002586 (DLC) 2010051563 (OCoLC)694393863 Encyclopedia of mathematics and its applications ; v. 141. http://id.loc.gov/authorities/names/n42010632 505-00/(S Cover -- Half Title -- Series Page -- Title -- Copyright -- Contents -- Preface -- 1 Asymptotics of Laplace-type integrals -- 1.1 Historical introduction -- 1.2 The method of steepest descents -- 1.2.1 Preliminaries -- 1.2.2 Asymptotic expansion of I (λ) -- 1.2.3 Quadratic and linear endpoint cases -- 1.2.4 Watson's lemma -- 1.2.5 Approximate methods -- The saddle-point method -- The method of stationary phase -- 1.3 Examples -- 1.4 Further examples -- 1.5 Uniform expansions -- 1.5.1 Saddle point near a pole -- 1.5.2 Saddle point near an endpoint -- 1.5.3 Two coalescing saddle points -- 1.6 Optimal truncation and superasymptotics -- 1.6.1 Optimal truncation -- 1.6.2 Ursell's lemma -- 1.7 The Stokes phenomenon -- 1.7.1 Description of the Stokes phenomenon -- 1.7.2 A rigorous approach -- 1.7.3 The Stokes phenomenon and steepest descents -- 1.8 Hyperasymptotics -- 1.8.1 Overview -- 1.8.2 Airey's converging factors -- 1.8.3 Dingle's converging factors -- 1.8.4 A formal discussion of hyperasymptotics -- 1.8.5 Truncation schemes -- 1.8.6 Hyperasymptotics of Laplace-type integrals -- 2 Hadamard expansion of Laplace integrals -- 2.1 Introduction -- 2.2 The Hadamard series for Iν(x) -- 2.2.1 Derivation of the single-stage expansion -- 2.2.2 The modified Hadamard series -- 2.2.3 The Hadamard series for complex z -- 2.2.4 Multi-stage expansions for Iν(z) -- 2.2.5 The Stokes phenomenon -- 2.2.6 The Hadamard series for 1F1(a -- a + b -- z) -- 2.3 Rapidly convergent Hadamard series -- 2.4 Hadamard series on an infinite interval -- 2.4.1 Subdivision of the integration path -- 2.4.2 The Hadamard expansion -- 2.4.3 Choice of the Ωn -- 2.4.4 A numerical example -- 2.4.5 A computational simplification -- 2.5 Examples -- 2.6 Bounds on the tails of Hadamard series -- 3 Hadamard expansion of Laplace-type integrals -- 3.1 Introduction -- 3.2 Expansion schemes. 505-00/(S 3.2.1 Subdivision of the u-axis -- 3.2.2 Expansion Scheme A -- 3.2.3 Expansion Scheme B -- 3.3 Examples -- Hadamard expansion near the Stokes line -- 3.4 Coalescence problems -- 3.4.1 Expansion scheme for coalescence problems -- 3.5 Examples of coalescence -- 4 Applications -- 4.1 Introduction -- 4.2 The Bessel function Jν(νz) -- 4.2.1 The case z positive -- 4.2.2 Numerical results -- 4.2.3 The case x -̃ 1 -- 4.2.4 The case z complex -- 4.3 The Pearcey integral -- 4.3.1 The Hadamard expansion of Pe(x, y) -- 4.3.2 Computation in the neighbourhood of the cusps -- 4.4 The parabolic cylinder function -- 4.5 The expansion for log Г(z) -- Appendix A: Properties of P(a, z) -- Appendix B: Convergence of Hadamard series -- Appendix C: Connection with the exp-arc integrals -- References -- Index. |
| spellingShingle | Paris, R. B. (Richard Bruce), 1946- Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / Encyclopedia of mathematics and its applications ; Preface; 1. Asymptotics of Laplace-type integrals; 2. Hadamard expansion of Laplace integrals; 3. Hadamard expansion of Laplace-type integrals; 4. Applications. Integral equations Asymptotic theory. http://id.loc.gov/authorities/subjects/sh85067089 Asymptotic expansions. http://id.loc.gov/authorities/subjects/sh85009056 Integral equations Asymptotic theory. Asymptotic expansions. Équations intégrales Théorie asymptotique. Développements asymptotiques. MATHEMATICS Algebra Abstract. bisacsh MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Asymptotic expansions fast Integral equations Asymptotic theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85067089 http://id.loc.gov/authorities/subjects/sh85009056 |
| title | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / |
| title_auth | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / |
| title_exact_search | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / |
| title_full | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / R.B. Paris. |
| title_fullStr | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / R.B. Paris. |
| title_full_unstemmed | Hadamard Expansions and Hyperasymptotic Evaluation : an Extension of the Method of Steepest Descents / R.B. Paris. |
| title_short | Hadamard Expansions and Hyperasymptotic Evaluation : |
| title_sort | hadamard expansions and hyperasymptotic evaluation an extension of the method of steepest descents |
| title_sub | an Extension of the Method of Steepest Descents / |
| topic | Integral equations Asymptotic theory. http://id.loc.gov/authorities/subjects/sh85067089 Asymptotic expansions. http://id.loc.gov/authorities/subjects/sh85009056 Integral equations Asymptotic theory. Asymptotic expansions. Équations intégrales Théorie asymptotique. Développements asymptotiques. MATHEMATICS Algebra Abstract. bisacsh MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Asymptotic expansions fast Integral equations Asymptotic theory fast |
| topic_facet | Integral equations Asymptotic theory. Asymptotic expansions. Équations intégrales Théorie asymptotique. Développements asymptotiques. MATHEMATICS Algebra Abstract. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Asymptotic expansions Integral equations Asymptotic theory |
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