Elliptic functions revisited

TL;DR

This paper offers a unified approach to elliptic functions through contour integrals, leading to new Fourier series representations and factorizations that bridge Jacobi and Weierstrass formulations.

Contribution

It introduces a novel contour integral method that unifies and simplifies the understanding of elliptic functions in their Jacobi and Weierstrass forms.

Findings

Unified contour integral representation of elliptic functions

New Fourier series expansion for elliptic functions

Factorization formulas derived from the renormalization approach

Abstract

Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson, Vorono\"i or Circle formulae), we will find the entirety of the elliptic functions, proposed either in the shape of Jacobi or Weierstrass. But with one translation which appears in their natural form. What could seem a defect will lead us to a renormalisation of the elliptic functions making it possible to determine, in a rather simple way, a Fourier series representation and a factorization of these functions.