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Re: Polynomial problems  Solid Harmonics
Posted:
Jun 18, 1996 3:09 AM


Tommy Nordgren wrote:
> I have a set of orthogonal polynomials in x,y,z, which is > GramScmidt 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]>(1z^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: +6193802734 The University of Western Australia Fax: +6193801014 Nedlands WA 6907 paul@physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/Paul _________________________________________________________________



