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 _________________________________________________________________ ==== [MESSAGE SEPARATOR] ====