Functions of Kolosov-Muskhelishvili and Holomorphicity Theorems

It is well known (see, e. g. Muskhelishvili [5]) that the equations of plane elasticity can be reduced to the bi-harmonic equation for Airy’s stress function. Hence, two harmonic functions are sufficient to represent a general solution. This means that the general solution in complex variables is represented by two holomorphic functions of the complex argument z = x + iy. The classical Goursat formula gives the connection between the real Airy function and these two holomorphic functions. By taking displacements and stresses expressed in terms of Airy’s function and using Goursat’s formula, we obtain the displacements and stresses expressed in terms of these two complex functions. This idea was first employed by Kolosov [1] and comprehensively exploited by Muskhelishvili [5].