Integral formulas of the Hilbert, Poincaré-Bertrand, Schwarz and Poisson type for the β–analytic function theory

Abstract

In the present work we obtain some analogues of the Hilbert formulas on the unit circle and on the upper half-plane for the theory of solutions of a special case of the Beltrami equation in C to be referred as β-analytic functions. Furthermore, a Poincaré–Bertrand formula related to the β-Cauchy singular integral over a closed Jordan curve is derived and it is used to derive the corresponding Schwarz and Poisson formulas.