Notebook to plot the prolate Speroidal wave functions
wmn scaling factor comes from
Theory and computation of spheroidal wavefunctions
P E Falloon, P C Abbott and J BWang

From Optical Imaging and Spectroscopy (Wiley - OSA 2009)
By David J. Brady
www.opticalimaging.org
Figure 10.12, 13, and 14

n=Table[NIntegrate[SpheroidalPS[0,0,i,x]^2,{x,0,100}],{i,2,8}];

Plot[Table[SpheroidalPS[0,0,i+1,x]/Sqrt[2*n[[i]]],{i,1,7}],{x,0,8},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[Table[SpheroidalPS[0,0,i+1,x]/Sqrt[2*n[[i]]]+.5 (i-1),{i,1,7}],{x,0,8},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

np=Table[NIntegrate[SpheroidalPS[i,0,5,x]^2,{x,0,20}],{i,0,8}];

dplots=Table[Plot[SpheroidalPS[i,0,5,x]/Sqrt[2*np[[i+1]]],{x,0,8},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black],{i,0,8}]

Show[GraphicsArray[Partition[dplots,3]]]

Plot[SpheroidalPS[4,0,10,x],{x,0,30},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[7,0,10,x],{x,0,30},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[10,0,10,x],{x,0,30},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[5,0,4,x],{x,0,30},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

norm=NIntegrate[SpheroidalPS[16,0,2,x]^2,{x,0,200}]

Plot[SpheroidalPS[16,0,2,x]/Sqrt[2 norm],{x,0,100},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[5,0,10,x],{x,0,100},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[5,0,5,x],{x,0,100},PlotRange->All,Axes->False,Frame->True,PlotStyle->Black]

Plot[SpheroidalPS[5,0,2,x],{x,0,5}]