Project III 2019-20

Topics in Complex Analysis: Periodic Functions according to Eisenstein

Thanasis Bouganis and Jens Funke

Description

You all know the standard trigonometric functions like sine, cosine, (co)tangent and their extensions to the complex plane via their charcterization as converging power series. It is noteworthy that one derive all the analytic but also the geometric properties of sine and cosine from this.  

In this project we follow a different approach to the trigonometric functions outlined by Eisenstein in the 1840s. Namely, we begin with the partial fraction expansion of the cotangent

for real (or better complex) x. This can be shown in various ways which is already interesting in its own right. Note that it is immediate that the infinite series defines a periodic function. Furthermore, it is important that the convergence of the infinite series is uniform in compact subsets away from the integers. Hence the series defines a meromorphic function which can be differentiated termwise.

But Eisenstein goes the other way around by starting with the expression on the right hand side. He then developes all the properties of the cotangent inclduing its differential equation etc just using algebraic manipulations! From there one can then derive the known properties of all trigonometric functions! This is the goal of the project! 

Depending on progress and interest we also study the Eisenstein-Weierstrass approach to doubly-periodic functions, so-called elliptic functions.

Initial reading and also the main sources throughout will be

  • Elliptic Functions according to Eisenstein and Kronecker (Chapter 1); A. Weil (Springer 1976)
  • Theory of Complex Functions (Chapter 11, Section 4); R. Remmert (Springer 1998)

Prerequisites

  • Complex Analyis II
  • Algebra II

email: J Funke