Introduction to elliptic functions 1000-1M10EF
Starting from the works of Gauss, Cauchy, Abel, Jacobi, Eisenstein, Riemann, Weierstrass, Klein and Poincare, the theory of meromorphic functions of a complex variable has developed significantly and has direct links with analysis, differential equations, algebra, number theory, potential theory, geometry and topology.
It makes an interesting and important topic for study. Since the theory is very rich, we shall mainly concentrate on the analytic viewpoint in the course. Connections with other areas will be discussed as well. We shall start with comprehensive introduction to elliptic functions, which are doubly periodic functions. They arose from attempts to evaluate certain integrals associated with the formula for the circumference of an ellipse. They can be regarded as meromorphic functions on the torus. Moreover, elliptic functions are the rational functions of the Weierstrass function and its derivative, these two functions being related by a first order nonlinear ordinary differential equation. Other topics that could be covered include automorphic functions, applications to number theory. If time permits, other special functions like hypergeometric
functions and their confluences, Lame functions etc will be discussed.
Prerequisite courses: ordinary differential equations, complex analysis.
Type of course
Prerequisites (description)
Learning outcomes
The basic knowledge of the theory of elliptic functions, elliptic integrals and elliptic curves.
Assessment criteria
Written exam or a presentation of project
Bibliography
T. Ekedahl, One semester of elliptic curves (available online in BUW)
N. Akhiezer, Elements of the theory of elliptic functions
Additonal literature:
J. V. Armitage and F. Eberlein, Elliptic functions
G. Jones, D. Singermann Complex functions: an algebraic and geometric viewpoint.
Batemann, Erdelyi, Higher transcendental functions
T. Apostol, Modular functions and Dirichlet series in number theory.
and other books available online in BUW
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: