| Description |
1 online resource (xi, 328 pages) : illustrations |
| Series |
Moscow lectures ; volume 9 |
|
Moscow lectures ; v. 9.
|
| Contents |
Intro -- Preface -- Acknowledgements -- Contents -- Chapter 0 Introduction -- 0.1 What is an Elliptic Function in one Phrase? -- 0.2 What Properties Do Elliptic Functions Have? -- 0.3 What Use Are Elliptic Functions? -- Pendulum -- Skipping rope -- Soliton equations -- Solvable lattice models -- Arithmetic-geometric mean -- Formula for solving quintic equations -- 0.4 A Small Digression on Elliptic Curves -- 0.5 Structure of this book -- Part I Real Part -- Chapter 1 The Arc Length of Curves -- 1.1 The Arc Length of the Ellipse -- 1.2 The Lemniscate and its Arc Length |
|
Chapter 2 Classification of Elliptic Integrals -- 2.1 What is an Elliptic Integral? -- 2.2 Classification of Elliptic Integrals -- (I) Standardising ( ) -- (II) Standardising the elliptic integral -- 2.3 Real Elliptic Integrals. -- Chapter 3 Applications of Elliptic Integrals -- 3.1 The Arithmetic-Geometric Mean -- 3.2 Motion of a Simple Pendulum -- Chapter 4 Jacobi's Elliptic Functions on -- 4.1 Jacobi's Elliptic Functions -- 4.2 Properties of Jacobi's Elliptic Functions -- 4.2.1 Troika of Jacobi's elliptic functions -- 4.2.2 Derivatives -- 4.2.3 Addition formulae |
|
Chapter 5 Applications of Jacobi's Elliptic Functions -- 5.1 Motion of a Simple Pendulum -- 5.2 The Shape of a Skipping Rope -- 5.2.1 Derivation of the differential equation -- 5.2.2 Solution of the differential equation and an elliptic function -- 5.2.3 The variational method -- Part II Complex Part -- Chapter 6 Riemann Surfaces of Algebraic Functions -- 6.1 Riemann Surfaces of Algebraic Functions -- 6.1.1 What is the problem? -- 6.1.2 Then, what should we do? -- 6.1.3 Another construction -- 6.1.4 The Riemann surface 1 -- 2 -- 6.2 Analysis on Riemann Surfaces |
|
6.2.1 Integrals on Riemann surfaces -- 6.2.2 Homology groups (a very short crash course) -- 6.2.3 Periods of one-forms -- Chapter 7 Elliptic Curves -- 7.1 The Riemann Surface ( ) -- 7.2 Compactification and Elliptic Curves -- 7.2.1 Embedding of R into the projective plane -- 7.2.2 Another way. (Embedding into (2)) -- 7.3 The Shape of R -- Chapter 8 Complex Elliptic Integrals -- 8.1 Complex Elliptic Integrals of the First Kind -- 8.2 Complex Elliptic Integrals of the Second Kind -- 8.3 Complex Elliptic Integrals of the Third Kind -- Chapter 9 Mapping the Upper Half Plane to a Rectangle |
|
9.1 The Riemann Mapping Theorem -- 9.2 The Reflection Principle -- 9.3 Holomorphic Mapping from the Upper Half Plane to a Rectangle -- 9.4 Elliptic Integrals on an Elliptic Curve -- Chapter 10 The Abel-Jacobi Theorem -- 10.1 Statement of the Abel-Jacobi Theorem -- 10.2 Abelian Differentials and Meromorphic Functions on an Elliptic Curve -- 10.2.1 Abelian differentials of the first kind -- 10.2.2 Abelian differentials of the second/third kinds and meromorphic functions -- 10.2.3 Construction of special meromorphic functions and Abelian differentials -- 10.3 Surjectivity of (Jacobi's Theorem) |
| Summary |
This book gives a comprehensive introduction to those parts of the theory of elliptic integrals and elliptic functions which provide illuminating examples in complex analysis, but which are not often covered in regular university courses. These examples form prototypes of major ideas in modern mathematics and were a driving force of the subject in the eighteenth and nineteenth centuries. In addition to giving an account of the main topics of the theory, the book also describes many applications, both in mathematics and in physics. For the readers convenience, all necessary preliminaries on basic notions such as Riemann surfaces are explained to a level sufficient to read the book. For each notion a clear motivation is given for its study, answering the question Why do we consider such objects?, and the theory is developed in a natural way that mirrors its historical development (e.g., If there is such and such an object, then you would surely expect this one). This feature sets this text apart from other books on the same theme, which are usually presented in a different order. Throughout, the concepts are augmented and clarified by numerous illustrations. Suitable for undergraduate and graduate students of mathematics, the book will also be of interest to researchers who are not familiar with elliptic functions and integrals, as well as math enthusiasts. |
| Bibliography |
Includes bibliographical references and index |
| Notes |
10.3.1 Finding an inverse image |
|
Description based on online resource; title from digital title page (viewed on October 13, 2023) |
| Subject |
Elliptic functions.
|
|
Elliptic functions
|
| Form |
Electronic book
|
| ISBN |
9783031302657 |
|
3031302656 |
|
