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IEEE Transactions on Antennas and Propagation
Volume 47 Number 1, January 1999

On a New Cylindrical Harmonic Representation for Spherical Waves

Page 97.

Abstract:

An exact series representation is presented for integrals whose integrands are products of cosine and spherical wave functions, where the argument of the cosine term can be any integral multiple n of the azimuth angle \phi. This series expansion will be shown to have the following form: I(n) = {{e^{-jkR_0}}\over {R_0}}\; \delta_{no} - jk\; \sum^{\infty}_{m=1}\; C(m, n)\; {{(k^2\rho\rho_0)}\over {m!}}\; {{h^{(2)}_m(kR_0)}\over {(kR_0)^m}}.It is demonstrated that in the special cases n = 0 and n = 1, this series representation corresponds to existing expressions for the cylindrical wire kernel and the uniform current circular loop vector potential, respectively. A new series representation for spherical waves in terms of cylindrical harmonics is then derived using this general series representation. Finally, a closed-form far-field approximation is developed and is shown to reduce to existing expressions for the cylindrical wire kernel and the uniform current loop vector potential as special cases.

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