Elliptic Integrals and Elliptic Functions 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. . 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.