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. [ABSTRACT FROM AUTHOR]