Two Simple Proofs of, Villat’s Formula and its Generalization in n-Connected Circular Domains

Abstract

This paper gives, in very simple terms, two new proofs of Villat’s formula of analytic functions in an annulus,by the properties of Schwarz integral and Poisson integral and the following representations of kernel functions:for q<|z|<1.Then the formula is generalized into n-Connected Circular domains. The final result took its concrete form and were read in 1964, earlierthan Dunducenho’s which was announced in MR 34# 6047,

Complete circle; Villat; Boundary area; Analytical function; Mathematical review; Schwarz; Poisson; Connected region; Integral representation; Imaginary axis;