Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

TL;DR

This paper introduces a novel Fourier decomposition method for the 3D Helmholtz Green function in cylindrical coordinates, providing closed-form solutions, differential equations, series expansions, and numerical comparisons to improve computational approaches.

Contribution

It presents a new Fourier decomposition technique with closed-form hypergeometric solutions and differential equations for the Helmholtz Green function in cylindrical coordinates, advancing analytical and numerical methods.

Findings

Closed-form solutions using Horn and Kampe de Feriet functions

Derivation of equivalent fourth-order differential equations

Numerical comparisons of Fourier coefficient calculation methods

Abstract

A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A second fourth-order ordinary differential equation for the general Fourier coefficent is derived from an integral representation of the coefficient, and both differential equations are shown to be equivalent. Series solutions for the various Fourier coefficients are also given, mostly in terms of Legendre functions and Bessel/Hankel functions. These are derived from the closed form hypergeometric solutions or an integral representation, or both. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented.