The Green's function on rhombic flat tori

Abstract

The Green's function $G$ is one of the single most powerful resources to solve Partial Differential Equations, specially those of Electromagnetism, Acoustics, Hydrodynamics and Particle Physics, among others. Although most of its applications occur in Physics, this function is an important mathematical tool, and in many cases it can be studied exclusively with methods in Geometry and Complex Analysis. In this master's dissertation we present a method to reliably and accurately obtain the Green's function $G$ for any flat rhombic torus $T$, as well as the necessary mathematical theory of elliptic functions to understand said method, in a computationally efficient manner, always with numerical values of significant digits up to the fourth decimal place (noting that $G$ is unique for $|T|=1$ and $\int_TGdA=0$). This precision is guaranteed by the strategies we adopt, which include theorems such as the Legendre Relation, properties of the Weierstra\ss\, P-Function, the algorithmic control of numerical errors as well as our methodology which forgoes the use of infinite series in favor of symmetric elliptic functions. Our code uses complex integration routines developed by H. Karcher, who also introduced the symmetric P-Weierstra\ss\,function, and these resources simplify the computation of elliptic functions considerably. At the end we present a section detailing how to adapt our methodology for finding $G$ without using the restriction $|T|=1$, as well as explaining how to use our code to compute $G$ and check its accuracy.