Painlevé Differential Equations in the Complex Plane
This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution o...
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Germany
De Gruyter
2008
Walter de Gruyter Walter de Gruyter GmbH |
| Vydání: | 1 |
| Edice: | De Gruyter Studies in Mathematics |
| Témata: | |
| ISBN: | 3110198096, 9783110198096, 9783110173796, 3110173794 |
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| Abstract | This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations. |
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| AbstractList | This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations. The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. |
| Author | Laine, Ilpo Shimomura, Shun Gromak, Valerii I |
| Author_FL | シモムラ, シュン |
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| Edition | 1 Reprint 2013. |
| Editor | Laine, Ilpo Shimomura, Shun |
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| Keywords | Komplexe Funktion Painlevé-Gleichung |
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| Notes | Includes bibliographical reference (p. [283]-299) and index |
| OCLC | 935264337 |
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| PageCount | 311 pages |
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| Snippet | This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the... The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from... |
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| SubjectTerms | Functions of complex variables Komplexe Funktion MATHEMATICS MATHEMATICS / General MATHEMATICS / Mathematical Analysis Painlevé equations Painlevé-Gleichung |
| TableOfContents | 35 Rational and classical transcendental solutions of (P3) when γδ ≠ 0 -- 8 The fifth Painlevé equation (P5) -- 36 Preliminary observations -- 37 The behavior of solutions near z = 0 and z = ∞ -- 38 The special case δ = 0, γ ≠ 0 -- 39 The Bäcklund transformations of (P5) -- 40 Rational and one-parameter families of solutions -- 41 Connection between (P3) and (P5) revisited -- 9 The sixth Painlevé equation (P6) -- 42 General properties of solutions -- 43 Pairs of differential equations equivalent to (P6) -- 44 A Riccati differential equation related to (P6) -- 45 A first order algebraic differential equation related to (P6) -- 46 Singular points of solutions of (P6) -- 47 Connection formulae between solutions of (P6) -- 48 Elementary solutions of (P6) -- 10 Applications of Painlevé equations -- 49 Partial differential equations related to Painlevé equations -- 50 Discrete Painlevé equations -- Appendix A Local existence and uniqueness of solutions of complex differential equations -- Appendix B Basic notations and facts in the Nevanlinna theory -- Bibliography -- Index Intro -- Introduction -- 1 Meromorphic nature of solutions -- 1 The first Painlevé equation (P1) -- 2 The second Painlevé equation (P2) -- 3 The third Painlevé equation (P3) -- 4 The fourth Painlevé equation (P4) -- 5 The fifth Painlevé equation (P5) -- 6 The sixth Painlevé equation (Pe) -- 2 Growth of Painlevé transcendents -- 7 Growth of first Painlevé transcendents -- 8 Growth of second and fourth Painlevé transcendents -- 9 Growth of third and fifth Painlevé transcendents -- 3 Value distribution of Painlevé transcendents -- 10 Deficiencies and ramification indices -- 11 The second main theorem for Painlevé transcendents -- 12 Value distribution with respect to small target functions -- 4 The first Painlevé equation (P1) -- 13 Nonexistence of the first integrals -- 14 Representation of solutions as quotient of entire functions -- 15 Special expansions of solutions -- 16 Higher order analogues of (P1) -- 5 The second Painlevé equation (P2) -- 17 Canonical representation of solutions -- 18 Poles of second Painlevé transcendents -- 19 The Bäcklund transformations of (P2) -- 20 Rational solutions of (P2) -- 21 The Airy solutions of (P2) -- 22 Higher order analogues of (P2) -- 6 The fourth Painlevé equation (P4) -- 23 Preliminary remarks -- 24 Poles of fourth Painlevé transcendents -- 25 Connection formulae between solutions of (P4) -- 26 Rational solutions of (P4) -- 27 The complementary error function hierarchy -- 28 A half-integer hierarchy of solutions -- 7 The third Painlevé equation (P3) -- 29 Preliminary remarks -- 30 Behavior of solutions around z = 0 and z = ∞ -- 31 Poles of third Painlevé transcendents -- 32 Canonical representation of solutions -- 33 The special case γ = 0, αδ ≠ 0 -- 34 Connection between solutions of (P3) and (P5) Chapter 5. The second Painlevé equation (P2) -- Contents -- Chapter 2. Growth of Painlevé transcendents -- Chapter 3. Value distribution of Painlevé transcendents -- Chapter 1. Meromorphic nature of solutions -- Chapter 6. The fourth Painlevé equation (P4) -- Chapter 10. Applications of Painlevé equations -- Appendix B. Basic notations and facts in the Nevanlinna theory -- Appendix A. Local existence and uniqueness of solutions of complex differential equations -- Chapter 8. The fifth Painlevé equation (P5) -- Backmatter Chapter 4. The first Painlevé equation (P1) -- Chapter 7. The third Painlevé equation (P3) -- Frontmatter -- Chapter 9. The sixth Painlevé equation (P6) -- |
| Title | Painlevé Differential Equations in the Complex Plane |
| URI | http://digital.casalini.it/9783110198096 https://cir.nii.ac.jp/crid/1130282272269912064 https://www.perlego.com/book/651405/painlev-differential-equations-in-the-complex-plane-pdf https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=314057 https://www.degruyterbrill.com/isbn/9783110198096 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9783110198096&uid=none |
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