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## Generalized Prolate Spheroids

 From: Carl Scarrott Subject: Generalized Prolate Spheroids Date: Tue, 12 Dec 2000 09:18:52 +0000

```Has anyone developed code for calculating the Generalized Prolate
Spheroidal functions?

They are found from the eigensolution of the integral equation:

lambda*phi(x)=Üint^{1}_{0} J_N(c*x*y) * sqrt(c*x*y) * phi(y) dy

for N=0,1,2,.... where J_N is the Bessel J function of order N, phi(x)
is the eigenfunction and lambda is the corresponding eigenvalue.

They are also the solution of the differential equation:

d/dx{ (1-x^2) * d/dx{phi(x)} } + (chi - (c*x)^2 + (0.25-N^2)/(x^2)) *
phi(x) = 0

where chi is the corresponding eigenvalue.

See Slepian (1964) Bell Sys. Tech. J. 44.

--
Carl Scarrott, BSc, GIMA  web   : http://www.maths.lancs.ac.uk/þscarrott
Maths & Stats Dept,       email : address@hidden
Lancaster University      Tel   : +44 (0)1524 594145
Lancaster, LA1 4YF, UK.   Fax   : +44 (0)1524 592681

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