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MathGroup Archive 1996

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Re: Polynomial problems - Solid Harmonics

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4269] Re: Polynomial problems - Solid Harmonics
  • From: Paul Abbott <paul at earwax.pd.uwa.edu.au>
  • Date: Sun, 23 Jun 1996 03:11:08 -0400
  • Organization: University of Western Australia
  • Sender: owner-wri-mathgroup at wolfram.com

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 at physics.uwa.edu.au 
AUSTRALIA                           http://www.pd.uwa.edu.au/Paul
_________________________________________________________________

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