Abstract
Rotational-translational addition theorems for the scalar spheroidal wave function ψ m n ( i ) ( h ; η , ξ , ϕ ) \psi _{mn}^{\left ( i \right )}\left ( {h;\eta ,\xi ,\phi } \right ) , with i = 1 , 3 , 4 i = 1,3,4 , are deduced. This permits one to represent the m n t h m{n^{th}} scalar spheroidal wave function, associated with one spheroidal coordinate system ( h q ; η q , ξ q , ϕ q ) \left ( {{h_q};{\eta _q},{\xi _q},{\phi _q}} \right ) centered at its local origin O q {O_q} , by an addition series of spheroidal wave functions associated with a second rotated and translated system ( h r ; η r , ξ r , ϕ r ) \left ( {{h_r};{\eta _r},{\xi _r},{\phi _r}} \right ) , centered at O r {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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