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In applied mathematics, oblate spheroidal wave functions are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, Δ Φ + k 2 Φ = 0 {\displaystyle \Delta \Phi +k^{2}\Phi =0} , by the method of separation of variables, {\displaystyle } , with:
the solution Φ {\displaystyle \Phi } can be written as the product of a radial spheroidal wave function R m n {\displaystyle R_{mn}} and an angular spheroidal wave function S m n {\displaystyle S_{mn}} by e i m φ {\displaystyle e^{im\varphi }}. Here c = k d / 2 {\displaystyle c=kd/2} , with d {\displaystyle d} being the interfocal length of the elliptical cross section of the oblate spheroid.
The radial wave function R m n {\displaystyle R_{mn}} satisfies the linear ordinary differential equation:
The angular wave function satisfies the differential equation: