Analytical properties of generalized biaxially symnnetric potentials

Abstract

The classical solutions (GBASP) to the generalized biaxially symmetric potential equation subject to certain Cauchy data along the singular lines may be expanded as Fourier series in terms of the complete set of normalized harmonic polynomials

The coefficients of these expansions characterize the singularities and the zeros of GBASP. R. P. Gilbert utilized operators based on Gegenbauer's integral for Jacobi polynomials and the envelope method to obtain Hadamard and Mandelbrojt classifications of the singularities. We establish Caratheodory–Toeplitz and Schur classifications of the zeros by means of convexity arguments and operators based on Koornwinder's new LaPlace type integral for Jacobi polynomials. The generalized translation operator establishes these criteria for the generalized translation of real GBASP.