In applied mathematics, oblate spheroidal wave functions (like also prolate spheroidal wave functions and other related 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, ( ξ , η , φ ) {\displaystyle (\xi ,\eta ,\varphi )} , with:
z = ( d / 2 ) ξ η , {\displaystyle \ z=(d/2)\xi \eta ,}
x = ( d / 2 ) ( ξ 2 + 1 ) ( 1 − η 2 ) cos φ , {\displaystyle \ x=(d/2){\sqrt {(\xi ^{2}+1)(1-\eta ^{2})}}\cos \varphi ,}
y = ( d / 2 ) ( ξ 2 + 1 ) ( 1 − η 2 ) sin φ , {\displaystyle \ y=(d/2){\sqrt {(\xi ^{2}+1)(1-\eta ^{2})}}\sin \varphi ,}
ξ ≥ 0 and | η | ≤ 1. {\displaystyle \ \xi \geq 0{\text{ and }}|\eta |\leq 1.}
the solution Φ ( ξ , η , φ ) {\displaystyle \Phi (\xi ,\eta ,\varphi )} can be written as the product of a radial spheroidal wave function R m n ( − i c , i ξ ) {\displaystyle R_{mn}(-ic,i\xi )} and an angular spheroidal wave function S m n ( − i c , η ) {\displaystyle S_{mn}(-ic,\eta )} 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 ( − i c , i ξ ) {\displaystyle R_{mn}(-ic,i\xi )} satisfies the linear ordinary differential equation:
( ξ 2 + 1 ) d 2 R m n ( − i c , i ξ ) d ξ 2 + 2 ξ d R m n ( − i c , i ξ ) d ξ − ( λ m n ( c ) − c 2 ξ 2 − m 2 ξ 2 + 1 ) R m n ( − i c , i ξ ) = 0 {\displaystyle \ (\xi ^{2}+1){\frac {d^{2}R_{mn}(-ic,i\xi )}{d\xi ^{2}}}+2\xi {\frac {dR_{mn}(-ic,i\xi )}{d\xi }}-\left(\lambda _{mn}(c)-c^{2}\xi ^{2}-{\frac {m^{2}}{\xi ^{2}+1}}\right){R_{mn}(-ic,i\xi )}=0} . The angular wave function satisfies the differential equation:
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