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Re: problem with DSolve and spheroidal harmonics

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  • Subject: [mg128302] Re: problem with DSolve and spheroidal harmonics
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Sat, 6 Oct 2012 01:47:11 -0400 (EDT)
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Am 05.10.2012 08:40, schrieb Tom Dickens:
> All,
>
> I wonder if this is a bug. I'm working on a project involving
> spheroidal harmonics, and tried the following example from the Help:
>
> DSolve[(1 - x^2) y''[x] -  2 x y'[x] + (SpheroidalEigenvalue[n, m, c]
> + c^2 (1 - x^2) - m^2-(  1 - x^2)) y[x] == 0, y, x]
>
>   {{y -> Function[{x},  C[1] SpheroidalPS[n, m, c, x] + C[2]
> SpheroidalQS[n, m, c, x]]}}
>
> This shows (in the Help) that the defining differential equation is
> solved correctly. However, when I run it in either v7 or v8 I get a
> differential root object
>
> {{y->DifferentialRoot[Function[{-[FormalY],-[FormalX]},{-[FormalY][-[FormalX]]
> (1--[FormalX]^2-c^2+-[FormalX]^2
> c^2+m^2-SpheroidalEigenvalue[n,m,c])+2 -[FormalX]
> (-[FormalY]^-[Prime])[-[FormalX]]+(-1+-[FormalX]^2)
> (-[FormalY]^-[Prime]-[Prime])[-[FormalX]]==0,-[FormalY][0]==C[1],(-[FormalY]^-[Prime])[0]==C[2]}]]}}
>
> Any thoughts?

The corrected differential equation you are writing about is most probably

 > DSolve[(1 - x^2) y''[x] -  2 x y'[x] + (SpheroidalEigenvalue[n, m, c]
 > + c^2 (1 - x^2) - m^2/(  1 - x^2)) y[x] == 0, y, x]
......................^^^^..
which is  the azimutal part of the sphere laplacian for
a product ansatz

Y =e^(i m phi) y(theta)
x=cos(theta)


Lap Y = 1/sin(theta) d_theta sin(theta) d_theta  y0
         + 1/sin(theta)^2 d_phi^2 y

-- 

Roland Franzius



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