Special Functions and Symbolic Summation

Wintersemester 2021

sradu@risc.jku.at

First lecture:

To be announced....

In this lecture we will introduce elliptic functions. In particular Weierstrass P function and Jacobi Theta functions will be studied and their main properties will be proven. Finally we introduce modular functions and modular forms and focus on the differential equations satisfied by viewing a modular form as a function of a modular function. Topics covered in the course are

Weierstrass P function and the Jacobi Theta functions
Riemann surfaces
Modular forms and modular functions
Yifan Yang's theorem on differential equations satisfied by modular forms
Dedekind Eta quotients

A major emphasis of the lecture is to present the basic notions, to develop the basic ideas of the underlying algorithms and to put computer algebra into action for concrete examples.

The material does not follow any particular book, however the notions used in the course are standard and can be googled or searched in Wikipedia.

Requirements: Basic knowledge from analysis and linear algebra.

Exercises

Exercise instructor: Koustav Banerjee (kbanerjee@risc.jku.at)

The exercises will be stated in the lectures in the form of "homeworks".