A topological interpretation of complex analytic behaviors of algebraic functions

This thesis aims to establish a topological result on algebraic functions satisfying a particular class of trinomial algebraic equation defining a complex algebraic curve. For this purpose, in the first chapter, we review some basic concepts of special functions namely, gamma and beta functions with the necessary properties to be used in the upcoming chapters. In Chapter 2, we briefly recall Gauss' hypergeometric function, which can be considered as a base case for a more general study presented in the third chapter. In Chapter 3, the first four sections are devoted to a review of well-known topological concepts in the literature. Starting from Section 5, we derive the Mellin-Barnes integral representations of solutions to a trinomial algebraic equation. The derivation of the integral representations of solutions is presented with the aid of tools introduced in the first two chapters. Then, we investigate the analytic continuations of the solutions along specified loops around branching points of the algebraic functions in question. In the last section, with the aid of the integral representations of solutions, we formulate the braid monodromy of algebraic functions in terms of rational twists that yield a classical Artin braid representation. Our method gives a precise description of the braid monodromy of algebraic functions with the aid of Mellin-Barnes integral representations. Thanks to Mellin-Barnes integral representations it is possible to pursue the angular movements of algebraic functions and to describe them in terms of rational twists. This enables us to deduce the Galois group of the algebraic equation in question after relating twist description with Artin braid representation.