Anti-unification (computer science)

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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,
<math> \Delta \Phi + k^2 \Phi=0</math>, by the method of separation of variables, <math>(\xi,\eta,\phi)</math>, with:

:<math>\ x=(d/2) \xi \eta, </math>

:<math>\ y=(d/2) \sqrt{(\xi^2+1)(1-\eta^2)} \cos \phi, </math>

:<math>\ z=(d/2) \sqrt{(\xi^2+1)(1-\eta^2)} \sin \phi, </math>

:<math>\ \xi \ge 0 \text{ and } |\eta| \le 1. </math>

the solution <math>\Phi(\xi,\eta,\phi)</math> can be written as the product of a radial spheroidal wave function <math>R_{mn}(-i c,i \xi)</math> and an angular spheroidal wave function <math>S_{mn}(-i c,\eta)</math> by <math>e^{i m \phi}</math>. Here <math>c=kd/2</math>, with <math>d</math> being the interfocal length of the elliptical cross section of the [[oblate spheroid]].

The radial wave function <math>R_{mn}(-i c,i \xi)</math> satisfies the linear [[ordinary differential equation]]:

:<math>\ (\xi^2 +1) \frac{d^2 R_{mn}(-i c,i \xi)}{d \xi ^2} + 2\xi \frac{d R_{mn}(-i c,i \xi)}{d \xi} -\left(\lambda_{mn}(c) -c^2 \xi^2 -\frac{m^2}{\xi^2+1}\right) {R_{mn}(-i c,i \xi)} = 0 </math>.

The angular wave function satisfies the differential equation:

:<math>\ (1- \eta^2) \frac{d^2 S_{mn}(-i c,\eta)}{d \eta ^2} - 2\eta \frac{d S_{mn}(-i c,\eta)}{d \eta} +\left(\lambda_{mn}(c) +c^2 \eta^2 -\frac{m^2}{1- \eta^2}\right) {S_{mn}(-i c,\eta)} = 0 </math>.

It is the same differential equation as in the case of the radial wave function. However, the range of the radial coordinate <math>\xi</math> is different than that of the angular coordinate <math>\eta</math>.

The eigenvalue <math>\lambda_{mn}(-ic)</math> of this Sturm-Liouville differential equation is fixed by the requirement that <math>{S_{mn}(-ic,\eta)}</math> must be finite for <math> |\eta| = 1</math>.

For <math>c=0</math> these two differential equations reduce to the equations satisfied by the [[associated Legendre polynomials]]. For <math>c\ne 0</math>, the angular spheroidal wave functions can be expanded as a series of Legendre functions.

The differential equations given above for the oblate radial and angular wave functions can be obtained from the corresponding equations for the [[prolate spheroidal wave functions]] by the substitution of <math> -ic </math> for <math> c </math> and <math> i \xi </math> for <math> \xi </math>. The notation for the oblate spheroidal functions reflects this relationship.

There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and Stegun.<ref>. M. Abramowitz and I. Stegun. ''Handbook of Mathematical Functions'' [http://www.math.sfu.ca/~cbm/aands/page_751.htm pp. 751-759] (Dover, New York, 1972)</ref> Abramowitz and Stegun (and the present article) follow the notation of Flammer.<ref name =Flammer>C. Flammer. ''Spheroidal Wave Functions'' Stanford University Press, Stanford, CA, 1957</ref>

Originally, the spheroidal wave functions were introduced by C. Niven,<ref>C. Niven ''[http://gallica.bnf.fr/ark:/12148/bpt6k55976x.image.f135.tableDesMatieres.langEN on the conduction of heat in ellipsoids of revolution.]'' Philosophical transactions of the Royal Society of London, '''171''' p. 117 (1880)</ref> 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,<ref>M. J. O. Strutt. ''Lamesche, Mathieusche and Verdandte Funktionen in Physik und Technik'' Ergebn. Math. u. Grenzeb, '''1''', pp. 199-323, 1932</ref> Stratton et al.,<ref>J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbato. ''Spheroidal Wave Functions'' Wiley, New York, 1956</ref> Meixner and Schafke,<ref>J. Meixner and F. W. Schafke. ''Mathieusche Funktionen und Sphäroidfunktionen'' Springer-Verlag, Berlin, 1954</ref> and Flammer.<ref name = Flammer/>

Flammer<ref name = 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.,<ref>A. L. Van Buren, R. V. Baier, and S Hanish ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=3050792 A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives.]'' (1970)</ref> King and Van Buren,<ref>B. J. King and A. L. Van Buren ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=124589 A Fortran computer program for calculating the prolate and oblate spheroidal angle functions of the first kind and their first and second derivatives.]'' (1970)</ref> Baier et al.,<ref>R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King - [http://dx.doi.org/10.1121/1.1974857 Spheroidal wave functions: their use and evaluation] The Journal of the Acoustical Society of America, '''48''', pp. 102–102 (1970)</ref> Zhang and Jin,<ref>S. Zhang and J. Jin. ''Computation of Special Functions'', Wiley, New York, 1996</ref> and Thompson.<ref>W. J. Thomson [http://www.ece.nus.edu.sg/stfpage/elelilw/Software/00764220.pdf Spheroidal Wave functions] Computing in Science & Engineering p. 84, May–June 1999</ref> Van Buren has recently developed new methods for calculating oblate spheroidal wave functions that extend the ability to obtain numerical values to extremely wide parameter ranges. These results are based on earlier work on prolate spheroidal wave functions.<ref>A. L. Van Buren and J. E. Boisvert. ''Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives'', Quarterly of Applied Mathemathics '''60''', pp. 589-599, 2002</ref><ref>A. L. Van Buren and J. E. Boisvert. ''Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives'', Quarterly of Applied Mathematics '''62''', pp. 493-507, 2004</ref> 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,<ref name = Flammer/> Hanish et al.,<ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=5502362 Tables of radial spheroidal wave functions, volume 4, oblate, m = 0]'' (1970)</ref><ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=124633 Tables of radial spheroidal wave functions, volume 5, oblate, m = 1]'' (1970)</ref><ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=124634 Tables of radial spheroidal wave functions, volume 6, oblate, m = 2]'' (1970)</ref> and Van Buren et al.<ref>A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish. ''Tables of Angular Spheroidal Wave Functions, vol. 2, oblate, m = 0'', Naval Research Lab. Publication, U. S. Govt. Printing Office, 1975</ref>

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

== References ==

<references/>

== External links ==

* MathWorld [http://mathworld.wolfram.com/SpheroidalWaveFunction.html Spheroidal Wave functions]
* MathWorld [http://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html Prolate Spheroidal Wave Function]
* MathWorld [http://mathworld.wolfram.com/OblateSpheroidalWaveFunction.html Oblate Spheroidal Wave function]





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Anti-unification (computer science)