# Prolate spheroidal wave function

In mathematics, the Prolate spheroidal wave functions (PSWF) are a set of orthogonal bandlimited functions. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. Let $D$ denote the time truncation operator, such that $f(t)=Df(t)$ if and only if $f(t)$ is timelimited within $[-T/2;T/2]$ . Similarly, let $B$ denote an ideal low-pass filtering operator, such that $f(t)=Bf(t)$ if and only if $f(t)$ is bandlimited within $[-\Omega ;\Omega ]$ . The operator $BD$ turns out to be linear, bounded and self-adjoint. For $n=0,1,2,\ldots$ we denote with $\psi _{n}(c,t)$ the n-th eigenfunction, defined as

$\ BD\psi _{n}(c,t)={\frac {1}{2\pi }}\int _{-\Omega }^{\Omega }\left(\int _{-T/2}^{T/2}\psi _{n}(c,\tau )e^{-i\omega \tau }d\tau \right)e^{i\omega t}d\omega =\lambda _{n}(c)\psi _{n}(c,t),$ where $1>\lambda _{0}(c)>\lambda _{1}(c)>\ldots >0$ are the associated eigenvalues, and $2c=T\Omega$ is a constant. The timelimited functions $\{\psi _{n}(c,t)\}_{n=0}^{\infty }$ are the Prolate Spheroidal Wave Functions (PSWFs). Pioneering work in this area was performed by Slepian and Pollak, Landau and Pollak, and Slepian.

These functions are also encountered in a different context. When solving the Helmholtz equation, $\Delta \Phi +k^{2}\Phi =0$ , by the method of separation of variables in prolate spheroidal coordinates, $(\xi ,\eta ,\phi )$ , with:

$\ x=(d/2)\xi \eta ,$ $\ y=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\cos \phi ,$ $\ z=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\sin \phi ,$ $\ \xi >=1$ and $|\eta |<=1$ .

the solution $\Phi (\xi ,\eta ,\phi )$ can be written as the product of a radial spheroidal wave function $R_{mn}(c,\xi )$ and an angular spheroidal wave function $S_{mn}(c,\eta )$ by $e^{im\phi }$ . Here $c=kd/2$ , with $d$ being the interfocal distance of the elliptical cross section of the prolate spheroid.

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

$\ (\xi ^{2}-1){\frac {d^{2}R_{mn}(c,\xi )}{d\xi ^{2}}}+2\xi {\frac {dR_{mn}(c,\xi )}{d\xi }}-\left(\lambda _{mn}(c)-c^{2}\xi ^{2}+{\frac {m^{2}}{\xi ^{2}-1}}\right){R_{mn}(c,\xi )}=0$ The eigenvalue $\lambda _{mn}(c)$ of this Sturm-Liouville differential equation is fixed by the requirement that ${R_{mn}(c,\xi )}$ must be finite for $|\xi |\to 1_{+}$ .

The angular wave function satisfies the differential equation:

$\ (\eta ^{2}-1){\frac {d^{2}S_{mn}(c,\eta )}{d\eta ^{2}}}+2\eta {\frac {dS_{mn}(c,\eta )}{d\eta }}-\left(\lambda _{mn}(c)-c^{2}\eta ^{2}+{\frac {m^{2}}{\eta ^{2}-1}}\right){S_{mn}(c,\eta )}=0$ It is the same differential equation as in the case of the radial wave function. However, the range of the variable is different (in the radial wave function, $\xi >=1$ ) in the angular wave function $|\eta |<=1$ ).

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 $S_{mn}(c,\eta )=(1-\eta ^{2})^{m/2}Y_{mn}(c,\eta )$ , the function $Y_{mn}(c,\eta )$ satisfies the following linear ordinary differential equation:

$\ (1-\eta ^{2}){\frac {d^{2}Y_{mn}(c,\eta )}{d\eta ^{2}}}-2(m+1)\eta {\frac {dY_{mn}(c,\eta )}{d\eta }}+\left(c^{2}\eta ^{2}+m(m+1)-\lambda _{mn}(c)\right){Y_{mn}(c,\eta )}=0,$ which is known as the spheroidal wave equation. This auxiliary equation is used for instance by Stratton in his 1935 article.

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 prolate and the oblate case. Computer programs for this purpose have been developed by many, including King et al., Patz and Van Buren, Baier et al., Zhang and Jin, Thompson, and Falloon. Van Buren and Boisvert have recently developed new methods for calculating prolate spheroidal wave functions that extend the ability to obtain numerical values to extremely wide parameter ranges. Fortran source code that combines the new results with traditional methods is available at http://www.mathieuandspheroidalwavefunctions.com.

Tables of numerical values of spheroidal wave functions are given in Flammer, Hunter, 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.

Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions" (see also Spectral concentration problem). These are of great utility in disciplines such as geodesy or cosmology.

Asymptotic expansions of angular prolate spheroidal wave functions for large values of $c$ have been derived by Müller. The relation between asymptotic expansions of spheroidal wave functions and Legendre functions has been investigated in Müller  and Müller.