# Geometric properties of special functions and related quadratic differentials

## Abstract

This dissertation introduces a new approach to view special functions from a quadratic differentials point of view. This idea allows us to study geometric properties of special functions. More specifically, this dissertation introduces quadratic differentials which are related to special functions, especially those related to theWeierstrass ℘ function, and discusses monotonicity properties, trajectory structures and domain configuration of the℘function on different shapes of the periodic cells. Additionally, it discusses some topological structure of the periodic cell in the image plane using the knowledge of quadratic differentials.In the first chapter, we discuss preliminary definitions and results for complex functions. If a complex valued function has an isolated singularity at a point z_0 ∈ C then it can be categorized as removable, pole or essential, by using series expansion of the function at z_0. If the series expansion have zero, finite, or infinite number of negative powers of z−z_0, then it is called removable, pole, or essential singularity respectively. Then important results that are related to this work is discussed.The second chapter introduces the our main definition in this dissertation, quadratic differential. It discusses local behaviour of critical points and domain configurations using Jenkins’ theorems and gives examples of such domains. In third chapter, elliptic functions are introduced. A function f(z) is said to be doubly periodic if it has two periods, w_1 and w_3, whose ratio w3/w1 is not real.A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function [22]. The w_1 and w_3 periods play the same part in the theory of elliptic functions as does the single period in the case of the trigonometric functions.Jacobi in 1835 [6] proved that if an univariate single-valued function is doubly periodic, then the ratio of periods cannot be real, as well as the impossibility fora single-valued univariate function to have more than two distinct periods [14].Then some of the properties of elliptic functions are discussed.The forth chapter introduces the most basic and famous function of elliptic function theory, the Weierstrass℘function. First, geometric properties of this function on periodic cell are discussed. Next, the concept of quadratic differential which we denoted by Q_℘(z)dz^2 on a periodic parallelogram using lemniscates of ℘ is introduced. Finally, the trajectory structure and domain configuration of this quadratic differential on different shapes of periodic parallelograms are identified.In ChapterV, another quadratic differential in the image plane of periodic cell which we denote by Q_w(w)dw_2 is introduced. Then the trajectory structure and domain configuration for this quadratic differential are identified. Next, the relation of the Weierstrass function to Tiechmuller’s problem is discussed. Finally, the renormalized Weierstrass function is introduced and discussed some geometric properties of elliptic modular function.

Lastly, in Chapter VI , the connection between Green’s function on the torus andWeierstrass function is obtained. The study of the Green’s function G on the torusT, that was initiated in [13], shows that the critical points of G are the solutions of the equation ζ(z) +a z+b ̄z= 0, where a and b are constants. From this equation we obtain a quadratic differential.