Beijing Institute of Mathematical Sciences and Applications

(This is a Qiuzhen college course.)

An elliptic function is defined as a doubly periodic meromorphic function on the complex plane. The study of elliptic integrals started by Fagnano, Euler, Legendre, Gauss and others, in the 18th century was turned into the theory of elliptic functions in the 19th century by Abel and Jacobi. Then Riemann, Weierstrass and Liouville developed the theory further by using complex analysis. The theory of elliptic functions and elliptic integrals thus founded is a prototype of today’s algebraic geometry. On the other hand elliptic functions and elliptic integrals appear in various problems in mathematics as well as in physics. Examples: arc length of an ellipse, arithmetic-geometric mean, solution of physical systems (pendulumn, top, skipping rope, the KdV equation), solution of quintic equations, etc. In this course we shall put emphasis on analytic aspects and applications.