Oblate spheroidal wave function

Oblate spheroidal wave functions are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, $\Delta \Phi +k^{2}\Phi =0$, by the method of separation of variables, $(\xi ,\eta ,\phi )$, with:

$\ x=f\xi \eta ,$

$\ y=f{\sqrt {(\xi ^{2}+1)(1-\eta ^{2})}}\cos \phi ,$

$\ z=f{\sqrt {(\xi ^{2}+1)(1-\eta ^{2})}}\sin \phi ,$

where ξ ≥ 0 and |η| ≤ 1, the solution $\Phi (\xi ,\eta ,\phi )$ can be written as the product of a radial spheroidal wave function $R_{mn}(-ic,i\xi )$ and an angular spheroidal wave function $S_{mn}(-ic,\eta )$ by $e^{im\phi }$ with $c=fk/2$.

The radial wave function $R_{mn}(-ic,i\xi )$ satisfies the linear ordinary differential equation:

$\ (\xi ^{2}-1){\frac {d^{2}R_{mn}(-ic,i\xi )}{d\xi ^{2}}}+2\xi {\frac {dR_{mn}(-ic,i\xi )}{d\xi }}-\left(\lambda _{mn}(c)-c^{2}\xi ^{2}+{\frac {m^{2}}{\xi ^{2}-1}}\right){R_{mn}(-ic,i\xi )}=0$

The angular wave function satisfies the differential equation:

$\ (\eta ^{2}-1){\frac {d^{2}S_{mn}(-ic,\eta )}{d\eta ^{2}}}+2\eta {\frac {dS_{mn}(-ic,\eta )}{d\eta }}-\left(\lambda _{mn}(c)+c^{2}\eta ^{2}-{\frac {m^{2}}{\eta ^{2}-1}}\right){S_{mn}(-ic,\eta )}=0$

The radial wave functions and the angular wave functions satisfy the same differential equation, although the range of the variable is different.

The eigenvalue $\lambda _{mn}(-ic)$ of this Sturm-Liouville differential equation is fixed by the requirement that ${S_{mn}(-ic,\eta )}$ must be finite for η = ± 1.

For $c=0$ these two differential equations reduce to the equations satisfied by the associated Legendre polynomials. For $c\neq 0$, the angular spheroidal wave functions can be expanded as a series of Legendre functions.

Let us note that if one writes $S_{mn}(-ic,\eta )=(1-\eta ^{2})^{m/2}Y_{mn}(-ic,\eta )$, the function $Y_{mn}(-ic,\eta )$ satisfies the following linear ordinary differential equation:

$\ (1-\eta ^{2}){\frac {d^{2}Y_{mn}(-ic,\eta )}{d\eta ^{2}}}-2(m+1)\eta {\frac {dY_{mn}(-ic,\eta )}{d\eta }}+\left(c^{2}\eta ^{2}+m(m+1)-\lambda _{mn}(c)\right){Y_{mn}(-ic,\eta )}=0,$

which is known as the spheroidal wave equation. This auxiliary equation is used for instance by Stratton in his 1935 article.

The corresponding differential equations for the prolate spheroidal wave functions are obtained from the oblate equations above by the substitution of c for -ic and ξ for iξ.

There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and Stegun . Abramowitz and Stegun (and the present article) follow the notation of Flammer.

Originally, the spheroidal wave functions were introduced by C. Niven, which lead to a Helmholtz equation in spheroidal coordinates. Monographs tying together many aspects of the theory of spheroidal wave functions were written by Strutt, Stratton et. al., Meixner and Schafke, and Flammer.

Flammer provided a thorough discussion of the calculation of the eigenvalues, angular wavefunctions, and radial wavefunctions for both the oblate and the prolate case. Computer programs for this purpose have been developed by many, including Van Buren et. al., King and Van Buren, Baier et. al., Zhang and Jin, and Thompson. Van Buren has recently developed new methods for calculating oblate spheroidal wave functions that extend the ablility to obtain numerical values to extremely wide parameter ranges. These results are based on earlier work on prolate spheroidal wave functions.. Fortran source code that combines the new results with traditional methods is available at http://www.mathieuandspheroidalwavefunctions.com.

Tables of numerical values of oblate spheroidal wave functions are given in Flammer, Hanish et. al., and Van Buren et. al..

The Digital Library of Mathematical Functions http://dlmf.nist.gov provided by NIST is an excellent resource for spheroidal wave functions.

External links

• MathWorld Spheroidal Wave functions
• MathWorld Prolate Spheroidal Wave Function
• MathWorld Oblate Spheroidal Wave function