Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function |
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Authors: | John T Conway and Howard S Cohl |
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Institution: | (1) Department of Mathematics, Macquarie University, Sydney, 2109, Australia;(2) Electronics Department, Gebze Institute of Technology, Gebze-Kocaeli, Turkey;(3) ICT Centre, CSIRO, Epping, NSW 1710, Australia |
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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 Green function
are split into their half advanced + half retarded and half advanced–half retarded components, and closed form solutions for
these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions
for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric
function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral
representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from
the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented.
Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly. |
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