Calculation of linear prolate spheroidal functions and eigen values

From Optical Imaging and Spectroscopy (Wiley - OSA 2009)
By David J. Brady
www.opticalimaging.org
Figure 10.15

As discussed in 

	W. P. Latham and M. L. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653- (1987)

the linear prolate spheroidal functions satisfy 

(Subscript[λ, n])(Subscript[ψ, n])(x)=(∫_(-1)^1)(e^icxy)(Subscript[ψ, n])(x)dx

This notebook finds the eigenvalues and wavefunctions for these functions using Gauss-Legendre integration

<<NumericalDifferentialEquationAnalysis`

orderN=100;
glw=GaussianQuadratureWeights[orderN,-1,1,12];

K combines the Fourier  kernel and the gl weights

Kw[c_]:=Table[Exp[ I Pi c glw[[i,1]] glw[[j,1]]/2]glw[[j,2]] ,{i,1,orderN},{j,1,orderN}];

Cp=20;
{lambda,phi}=Eigensystem[Kw[Cp]];
p20=ListLogPlot[Cp Abs[lambda]^2/4,Joined->True,Axes->False, Frame->True,PlotStyle->{Black},PlotRange->{{0,80},{10^(-6),2}},FrameLabel->{n,(Subscript[λ, n])}]

Cp=40;
{lambda,phi}=Eigensystem[Kw[Cp]];
p40=ListLogPlot[Cp Abs[lambda]^2/4,Joined->True,Axes->False, Frame->True,PlotStyle->{Black},PlotRange->{{0,80},{10^(-6),2}},FrameLabel->{n,(Subscript[λ, n])}]

Cp=60;
{lambda,phi}=Eigensystem[Kw[Cp]];
p60=ListLogPlot[Cp Abs[lambda]^2/4,Joined->True,Axes->False, Frame->True,PlotStyle->{Black},PlotRange->{{0,80},{10^(-6),2}},FrameLabel->{n,(Subscript[λ, n])}]

Show[{p20,p40,p60}]