Polyanalytic Functions and Their Generalizations

Abstract

Polyanalytic functions emerged in the mathematical theory of elasticity: eighty years after the discovery of its basic equations, Kolossoff found that functions of the form φ(z) + Ψ(z), where φ and Ψ are analytic functions, can be an efficient tool for solving problems of the planar theory of elasticity. Functions of this form were later called bianalytic. Useful applications of this idea in mechanics are widely known from the remarkable investigations by Kolossoff, Muskhelishvili and their followers. The class of polyanalytic functions is an extension of the class of bianalytic functions. This survey is devoted to the former class.