On Schwartz-Villat's formula for analytic functions

In this present study, we intend to obtain a new criterion for the analyticity of a complex function. More exactly, we will deduce some concrete restrictions that must be performed by a function that is complex, is defined on the disc centred at the origin that has radius 1 (hereinafter referred to as the unit disc), which, in addition, is continuous on this disc, including its border, to be analytic on the open unit disc. After that, as applications, we will deduce the Schwartz-Villat's formula in this context, and we will obtain the known Joukowski's function.

Automated summary

The aim of this study is to establish a new criterion for the analyticity of a complex function. Specifically, we will derive specific restrictions that a complex function defined on the unit disc must satisfy in order to be analytic on the open unit disc. We will then apply these findings to derive Schwartz-Villat's formula and obtain the well-known Joukowski's function.