在數學中，**長球波函數**由一個限時、限頻、與第二個限時的函數相乘而成。假定{\displaystyle Q\_{T}}表示一個切截時間的運算器，且{\displaystyle x=Q\_{T}x}，則x必為有限時間區間的函數，當x在{\displaystyle [-T/2;T/2]}的區間內。同理，假定{\displaystyle P\_{W}}表示一個理想的低頻濾波器，且{\displaystyle x=P\_{W}x}，則x必為有限頻寬區間的函數，當x在{\displaystyle [-W;W]}的區間內。透過組合上述運算子，使得{\displaystyle Q\_{T}P\_{W}Q\_{T}}轉變成線性、有界且自伴的運算式。對於{\displaystyle n=1,2,\ldots }，我們假設\psi\_n為第n項的本徵函數，定義下列函式

{\displaystyle \ Q\_{T}P\_{W}Q\_{T}\psi \_{n}=\lambda \_{n}\psi \_{n},}
其中{\displaystyle \{\lambda \_{n}\}\_{n}}為對應的本徵值。此時限函數{\displaystyle \{\psi \_{n}\}\_{n}}即為長球波函數(PSWFs).在此領域中，幾個非常重要的先驅文章由Slepian and Pollak, Landau and Pollak, and Slepian.所提出。

這些函數有些不同的意涵，當在解亥姆霍兹方程{\displaystyle \Delta \Phi +k^{2}\Phi =0}，透過在長球面坐標系做變數分離，使得{\displaystyle (\xi ,\eta ,\phi )}各代表：

{\displaystyle \ x=(d/2)\xi \eta ,}
{\displaystyle \ y=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\cos \phi ,}
{\displaystyle \ z=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\sin \phi ,}
{\displaystyle \ \xi >=1} and {\displaystyle |\eta |<=1}.
得到解{\displaystyle \Phi (\xi ,\eta ,\phi )}為長球波函數{\displaystyle R\_{mn}(c,\xi )}與角球波函數{\displaystyle S\_{mn}(c,\eta )}的成積乘上{\displaystyle e^{im\phi }}. 這裡的變數c可定義為{\displaystyle c=kd/2}, with d為長球的橢圓截面的兩焦點的距離。

徑向波(The radial wave function){\displaystyle R\_{mn}(c,\xi )}滿足線性常微分方程:

{\displaystyle \ (\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}
此本徵值{\displaystyle \lambda \_{mn}(c)}在Sturm-Liouville 微分方程中已被固定，透過設定{\displaystyle {R\_{mn}(c,\xi )}}為有限函數，當 {\displaystyle |\xi |\to 1\_{+}}.

角波函數滿足下列微分方程：

{\displaystyle \ (\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}
這跟徑向波函數式為同樣的微分方程。然而，這兩式的變數的範圍是不同的(在徑向波函數中{\displaystyle \xi >=1})，在角波函數中{\displaystyle |\eta |<=1})。

當給定c=0，這兩個微分方程可以簡化成滿足伴隨勒讓德多項式的式子。當給定{\displaystyle c\neq 0}，角波函數可被展開成勒讓德級數。

注意，如果我們將角波函數寫成{\displaystyle S\_{mn}(c,\eta )=(1-\eta ^{2})^{m/2}Y\_{mn}(c,\eta )}，函數{\displaystyle Y\_{mn}(c,\eta )}將滿足以下線性微分方程：

{\displaystyle \ (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,}

此函數為球波函數。這個輔助方程式在Stratton 1935年的文章被當作例子。

現存不少不同的球函數標準化的方法，在Abramowitz and Stegun.的文章中有整理的表格。Abramowitz跟Stegun(以及現在的相關文章)都沿用Flammer當初提出來的符號。

一開始，球波函數是由C. Niven,提出，他在球座標上引入Helmholtz方程式。許多專題論文已經探討出球波函數的很多面向，例如Strutt, Stratton et al. Meixner and Schafke, and Flammer.等人的作品。

Flammer提供了一個完整的討論，計算出長球與扁球的本徵值、角波函數與徑函數。許多電腦程式已經因應發展出來，其中包含King與其團隊, Patz和Van Buren, Baier與其團隊, Zhang和Jin, Thompson,、Falloon. Van Buren和Boisvert最近發展出新的方法去計算出長球波函數，延伸了數值解的能力，能運算極廣的變數範圍。Fortran原始碼結合了新的結果與傳統的方法，可見於http://www.mathieuandspheroidalwavefunctions.com.。 （页面存档备份，存于）

Flammer, Hunter, Hanish et al. and Van Buren et al.等人也提出了數值解的整理表格。

NIST提供的DLMF(Digital Library of Mathematical Functions)(

關於值域落在單位球的表面的長球波函數，我們通稱為"Slepian functions" (另見「頻譜集中問題」)。這函數存在非常多的應用，像是大地測量以及宇宙學.

參考文獻
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1. ↑  D. Slepian and H. O. Pollak. *Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I[永久]* Bell System Technical Journal **40** (1961)
2. ↑  H. J. Landau and H. O. Pollak. *Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II[永久]* Bell System Technical Journal **40** (1961)
3. ↑  H. J. Landau and H. O. Pollak. *Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty -- III: The Dimension of the Space of Essentially Time- and Band-Limited Signals[永久]* Bell System Technical Journal **41** (1962)
4. ↑  D. Slepian *Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions[永久]* Bell System Technical Journal **43** (1964)
5. ↑  D. Slepian. *Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case[永久]* Bell System Technical Journal **57** (1978)
6. ↑  J. A. Stratton *Spheroidal functions （页面存档备份，存于）* Proceedings of the National Academy of Sciences (USA) **21**, 51 (1935)
7. ↑ . M. Abramowitz and I. Stegun. *Handbook of Mathematical Functions* pp. 751-759 （页面存档备份，存于） (Dover, New York, 1972)
8. 1 2 3 4  C. Flammer. *Spheroidal Wave Functions* Stanford University Press, Stanford, CA, 1957
9. ↑  C. Niven *)On the conduction of heat in ellipsoids of revolution. （页面存档备份，存于）* Philosophical transactions of the Royal Society of London, **171** p. 117 (1880)
10. ↑  M. J. O. Strutt. *Lamesche, Mathieusche and Verdandte Funktionen in Physik und Technik* Ergebn. Math. u. Grenzeb, **1**, pp. 199-323, 1932
11. ↑  J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbato. *Spheroidal Wave Functions* Wiley, New York, 1956
12. ↑  J. Meixner and F. W. Schafke. *Mathieusche Funktionen und Sphäroidfunktionen* Springer-Verlag, Berlin, 1954
13. ↑  B. J. King, R. V. Baier, and S Hanish *A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. （页面存档备份，存于）* (1970)
14. ↑  B. J. Patz and A. L. Van Buren *A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind. （页面存档备份，存于）* (1981)
15. ↑  R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King - Spheroidal wave functions: their use and evaluation （页面存档备份，存于） The Journal of the Acoustical Society of America, **48**, pp. 102–102 (1970)
16. ↑  S. Zhang and J. Jin. *Computation of Special Functions*, Wiley, New York, 1996
17. ↑  W. J. Thomson Spheroidal Wave functions 的存檔，存档日期2010-02-16. Computing in Science & Engineering p. 84, May–June 1999
18. ↑  P. E. Falloon Thesis on numerical computation of spheroidal functions 的存檔，存档日期2011-04-11. University of Western Australia, 2002
19. ↑  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
20. ↑  A. L. Van Buren和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
21. ↑  H. E. Hunter *Tables of prolate spheroidal functions for m=0: Volume I. （页面存档备份，存于）* (1965)
22. ↑  H. E. Hunter *Tables of prolate spheroidal functions for m=0 : Volume II. （页面存档备份，存于）* (1965)
23. ↑  S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King *Tables of radial spheroidal wave functions, volume 1, prolate, m = 0* (1970)
24. ↑  S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King *Tables of radial spheroidal wave functions, volume 2, prolate, m = 1 （页面存档备份，存于）* (1970)
25. ↑  S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King *Tables of radial spheroidal wave functions, volume 3, prolate, m = 2 （页面存档备份，存于）* (1970)
26. ↑  A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish. *Tables of angular spheroidal wave functions, vol. 1, prolate, m = 0*, Naval Research Lab. Publication, U. S. Govt. Printing Office, 1975
27. ↑  F. J. Simons, M. A. Wieczorek and F. A. Dahlen. *Spatiospectral concentration on a sphere*. SIAM Review, 2006, doi:10.1137/S0036144504445765
28. ↑  F. J. Simons and Dahlen, F. A. *Spherical Slepian functions and the polar gap in Geodesy*. Geophysical Journal International, 2006, doi:10.1111/j.1365-246X.2006.03065.x
29. ↑  F. A. Dahlen和F. J. Simons. *Spectral estimation on a sphere in geophysics and cosmology*. Geophysical Journal International, 2008, doi:10.1111/j.1365-246X.2008.03854.x

外部連結
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* MathWorld Spheroidal Wave functions （页面存档备份，存于）
* MathWorld Prolate Spheroidal Wave Function （页面存档备份，存于）
* MathWorld Oblate Spheroidal Wave function （页面存档备份，存于）