Laplace Series

Author

Eric W. Weisstein
March 4, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/LaplaceSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LaplaceSeries.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Example

g[θ_,ϕ_]:=3+Cos[2θ]^3+Sin[ϕ]/2

Integrate over the complex conjugate

coef=Integrate[(-1)^m SphericalHarmonicY[l,-m,θ,ϕ]g[θ,ϕ]Sin[θ],{ϕ,0,2Pi},{θ,0,Pi}]

f=With[{n=10},
Table[SphericalHarmonicY[l,m,θ,ϕ],{l,0,n},{m,0,l}]
Table[coef,{l,0,n},{m,0,l}]];//Timing

f//ColumnForm//TraditionalForm

<<Utilities`Typesetting`
<<Graphics`ParametricPlot3D`

MakeBoxes[label,TraditionalForm]:=
RowBox[{RowBox[{"f","(",RowBox[{"θ",",","ϕ"}],")"}],"==",RowBox[{RowBox[{SuperscriptBox["cos","3"],"(",RowBox[{"2"," ","θ"}],")"}],"+",RowBox[{RowBox[{"(","sin","ϕ",")"}],"/","2"}],"+","3"}]}]

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
SphericalPlot3D[Evaluate[Abs[Tr[Flatten[Take[f,2]]]]],{θ,0,Pi},{ϕ,0,2Pi},Ticks->None,PlotLabel->TraditionalForm[l==2]],
SphericalPlot3D[Evaluate[Abs[Tr[Flatten[Take[f,5]]]]],{θ,0,Pi},{ϕ,0,2Pi},Ticks->None,PlotLabel->TraditionalForm[l==5]],
SphericalPlot3D[Evaluate[Abs[Tr[Flatten[Take[f,10]]]]],{θ,0,Pi},{ϕ,0,2Pi},Ticks->None,PlotLabel->TraditionalForm[l==10]],
SphericalPlot3D[3+Cos[2θ]^3+Sin[ϕ]/2,{θ,0,Pi},{ϕ,0,2Pi},Ticks->None,PlotLabel->TraditionalForm[label]]
}
]],PlotRange->All,ImageSize->500]