A Poisson integral formula for the ellipse and some applications

Abstract

that converges uniformly interior to E. Consequently, if U(x, y) is any function that is harmonic in E, then U can be represented as the real part of some f that is given by (1). We shall use this fact to establish a Poisson integral formula for U. We should mention here that the Poisson formula can also be obtained by mapping the ellipse in the z-plane onto a rectangle in the w-plane thus obtaining F(w) as the transform of f(z) and applying the Cauchy integral formula, using in addition some properties of the Jacobian elliptic function Z(w). We have chosen the method presented in this paper because of its simplicity. The Poisson formula is readily generalized to the Poisson-Stieltjes formula and to what is commonly called the Herglotz formula for nonnegative harmonic functions. We apply the Poisson-Stieltjes formula to the class 61 of functions with positive real part in E and extend it to find an integral representation for functions regular in E and mapping E onto a domain that is starlike with respect to the origin. Other classes of functions related to 61 in various ways could also be studied by this method. The coefficient problem for the class of starlike functions as well as other subclasses of functions that are univalent in E and can be related to 61 is studied in [4].