A representation of the solutions of the generalized Cauchy-Riemann equations and its applications

Abstract

A method of integral representations for the generalized Cauchy-Riemann system in terms of an arbitrary analytic function, similar to the well-known Whittaker-Polozhii representation /1/, is developed. The representation includes various well-known results as special cases, and the limiting case leads to the classical representation of the theory of a generalized axisymmetric potential. The representations established are used to reduce mixed problems for the system to paired equations and then to a Fredholm equation of the second kind. At the same time, a device is described for regularizing paired equations, and a case in which a closed solution exists is presented. The results are extended to a sytem of more-general form and also to second-order equations, whose type and dimensionality are not essential. It is shown that the integral operators constructed here convert the solution of a parabolic or hyperbolic equation with variable coefficients into a solution of the classical equations of heat conduction and wave propagation, thus furnishing an explicit representation for solutions of the corresponding Cauchy problems. The effectiveness of the approach is demonstrated with reference to the problem of inflow in a fissure in an inhomogeneous layer of finite width. Simple formulae for the pressure and discharge of fissures are presented.