Rotational-translational addition theorems for scalar spheroidal wave functions

MacPhie, R. H.; Dalmas, J.; Deleuil, R.

doi:10.1090/qam/872824

Authors: R. H. MacPhie, J. Dalmas and R. Deleuil
Journal: Quart. Appl. Math. 44 (1987), 737-749
MSC: Primary 33A55
DOI: https://doi.org/10.1090/qam/872824
MathSciNet review: 872824
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Abstract: Rotational-translational addition theorems for the scalar spheroidal wave function $\psi _{mn}^{\left ( i \right )}\left ( {h;\eta ,\xi ,\phi } \right )$, with $i = 1,3,4$, are deduced. This permits one to represent the $m{n^{th}}$ scalar spheroidal wave function, associated with one spheroidal coordinate system $\left ( {{h_q};{\eta _q},{\xi _q},{\phi _q}} \right )$ centered at its local origin ${O_q}$, by an addition series of spheroidal wave functions associated with a second rotated and translated system $\left ( {{h_r};{\eta _r},{\xi _r},{\phi _r}} \right )$, centered at ${O_r}$. Such theorems are necessary in the rigorous analysis of radiation and scattering by spheroids with arbitrary spacings and orientations.

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