Quasi-translational addition expressions of prolate scalar wave functions for thin prolate spheroids

The conventional, translational addition theorems for both spherical and prolate spheroidal wave functions applied to incoming or outgoing waves possess a well-known convergence sphere near which the addition series fail to converge rapidly. To overcome this deficiency the quasi-translational addition expressions for prolate wave functions are developed by using a physical-geometrical transformation which allows the separation in the azimuthal direction in the translated prolate spheroidal coordinate system. These quasi-translational addition expressions are valid everywhere and, in particular, exhibit a fast convergence characteristics for thin spheroids which are commonly used in physical and engineering applications