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＜E-Book＞
Elliptic Integrals and Elliptic Functions / by Takashi Takebe
(Moscow Lectures. ISSN:25220322 ; 9)

Edition	1st ed. 2023.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2023
Language	English
Size	XI, 328 p. 90 illus., 16 illus. in color : online resource
Authors	*Takebe, Takashi author SpringerLink (Online service)
Subjects	LCSH:Special functions LCSH:Functions of complex variables LCSH:Mathematical physics FREE:Special Functions FREE:Functions of a Complex Variable FREE:Mathematical Methods in Physics
Notes	Introduction -- Chapter 1. The arc length of curves -- Chapter 2. Classification of elliptic integrals -- Chapter 3. Applications of elliptic integrals -- Chapter 4. Jacobi’s elliptic functions on R -- Chapter 5. Applications of Jacobi’s elliptic functions -- Riemann surfaces of algebraic functions -- Chapter 7. Elliptic curves -- Chapter 8. Complex elliptic integrals -- Chapter 9. Mapping the upper half plane to a rectangle -- Chapter 10. The Abel-Jacobi theorem -- Chapter 11. The general theory of elliptic functions -- Chapter 12. The Weierstrass ℘-function -- Chapter 13. Addition theorems -- Chapter 14. Characterisation by addition formulae -- Chapter 15. Theta functions -- Chapter 16. Infinite product factorisation of theta functions -- Chapter 17. Complex Jacobian functions -- Appendix A. Theorems in analysis and complex analysis -- Bibliography -- Index 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 reader’s 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. HTTP:URL=https://doi.org/10.1007/978-3-031-30265-7

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