Spheroidal Wave Function

Author

Eric W. Weisstein
October 10, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SpheroidalWaveFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SpheroidalWaveFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

SpheroidalS1

Notation

SpheroidalS1[n,m,γ,x]//TraditionalForm

Normalization

Integrate[((SpheroidalS1[n,m,γ,x])^2),{x,-1,1}]

Asymptotic

Series[SpheroidalS1[n,m,γ,x],{x,∞,3}]

Derivative

D[SpheroidalS1[n,m,γ,x],x]

Special Cases

Table[SpheroidalS1[n,1,n (π/2),x],{n,4}]//FullSimplify

ODE

Something's wrong here

With[{n=1,m=2,γ=1.23},NIntegrate[((SpheroidalS1[n,m,γ,x])^2),{x,-1,1}]
]

DSolve[ode=D[(1-(η^2))S'[η]]+((Subscript[λ, mn])-(c^2)(η^2)-((m^2)/(1-(η^2))))S[η]==0,S[η],η]

soln=FullSimplify[%][[1,1,2]]

Simplify[ode/.S->Function[η,Evaluate[soln]]]

SpheroidalS2

Notation

SpheroidalS2[n,m,γ,x]//TraditionalForm

Normalization

Integrate[((SpheroidalS2[n,m,γ,x])^2),{x,-1,1}]

Derivative

D[SpheroidalS2[n,m,γ,x],x]

Asymptotic

Series[SpheroidalS2[n,m,γ,x],{x,∞,3}]

Special Cases

Table[SpheroidalS2[n,1,n (π/2),x],{n,4}]//FullSimplify

SpheroidalPS

Notation

SpheroidalPS[n,m,γ,x]//TraditionalForm

Series

Series[SpheroidalPS[n,m,γ,x],{γ,0,3}]//TraditionalForm

Series[SpheroidalPS[n,m,γ,x],{x,0,2}]

Derivative

D[SpheroidalPS[n,m,γ,x],x]

Special Cases

Table[SpheroidalPS[n,1,n (π/2),x],{n,4}]//FullSimplify

SpheroidalQS

Notation

SpheroidalQS[n,m,γ,x]//TraditionalForm

Series

Series[SpheroidalQS[n,m,γ,x],{γ,0,3}]//TraditionalForm

Series[SpheroidalQS[n,m,γ,x],{x,0,2}]

Derivative

D[SpheroidalQS[n,m,γ,x],x]

Special Cases

Table[SpheroidalQS[n,1,n (π/2),x],{n,4}]//FullSimplify