Saad Abed: Gamma function related to Pick function

Abstract:

This thesis is divided into two parts. In the first part we will study some properties of the Gamma function, Γ(z), which can be viewed as an extension of the factorial function (n + 1) 7→ n! to a subset of the complex plane (more precisely to C / Z≤0). Γ(x) can be given by several representations, and we will offer some of them, The second part of this thesis is about Pick functions and mainly follows the paper Pick Functions Related to the Gamma Function by C. Berg and H.L. Pedersen (see [BP]). Pick functions are holomorphic functions from the open upper complex half plane to the closed upper complex half plane. We will prove that a special class of maps are Pick functions. We end up with proving that Log(Γ(z + 1))/z is a Pick function which connects the two subjects of my thesis.