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Topic: Re: Polynomial problems - Solid Harmonics
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Paul Abbott

Posts: 2
Registered: 12/7/04
Re: Polynomial problems - Solid Harmonics
Posted: Jun 18, 1996 3:09 AM
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Tommy Nordgren wrote:

> I have a set of orthogonal polynomials in x,y,z, which is
> Gram-Scmidt orthogonalized with respect to integration over the unit
> sphere.


Aren't you are working with the Solid Harmonics then?

The solid harmonics are closely related to the Spherical Harmonics.
After defining a suitable transformation between spherical polar
coordinates and cartesian coordinates:

rtpToxyz = {Exp[Complex[0,n_] p] ->
((x+Sign[n] I y)/(r Sin[t]))^Abs[n],
Cos[t]->z/r,
Sin[z]->(1-z^2/r^2)^(1/2)};

the (complex) solid harmonics are:

SolidHarmonics[l_,m_,x_,y_,z_] :=
((r^l SphericalHarmonicY[l,m,t,p] /.
rtpToxyz) /. r->(x^2+y^2+z^2)^(1/2)) // Simplify

For example,

SolidHarmonics[2,1,x,y,z] // ComplexExpand

5
-3 Sqrt[----] x z
6 Pi 3 I 5
----------------- - --- Sqrt[----] y z
2 2 6 Pi

The solid harmonics are orthonormal and very easily computed.

Cheers,
Paul

_________________________________________________________________
Paul Abbott
Department of Physics Phone: +61-9-380-2734
The University of Western Australia Fax: +61-9-380-1014
Nedlands WA 6907 paul@physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
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