Spherical Harmonic Addition Theorem

Author

Eric W. Weisstein
December 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SphericalHarmonicAdditionTheorem.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sums

((4 π)/(2 l+1))(∑_(m=-l)^l)(((-1))^m) (Subscript[Y, l]^m)(θ1,ϕ1) (Subscript[Y, l]^(-m))(θ2,ϕ2)==(LegendreP[l, sin(θ1) sin(θ2) cos(ϕ1-ϕ2)+cos(θ1) cos(θ2)])

((4π)/(2l+1))Sum[(((-1))^m)SphericalHarmonicY[l,m,θ1,ϕ1]SphericalHarmonicY[l,-m,θ2,ϕ2],{m,-l,l}]

Union[Chop[Apply[Subtract, 
    Table[{(4*Pi*Sum[(-1)^m*SphericalHarmonicY[l, -m, θ2, ϕ2]*SphericalHarmonicY[l, m, θ1, ϕ1], 
          {m, -l, l}])/(1 + 2*l), LegendreP[l, Cos[θ1]*Cos[θ2] + Sin[θ1]*Sin[θ2]*Cos[ϕ1 - ϕ2]]} /. 
      {l -> RandomInteger[{0, 10}], θ1 -> RandomReal[{0, 2*Pi}], ϕ1 -> RandomReal[{0, Pi}], 
       θ2 -> RandomReal[{0, 2*Pi}], ϕ2 -> RandomReal[{0, Pi}]}, {100}], {1}]]]

(LegendreP[l, sin(θ1) sin(θ2) cos(ϕ1-ϕ2)+cos(θ1) cos(θ2)])==(LegendreP[l, cos(θ1)]) (LegendreP[l, cos(θ2)])+2 (∑_m^l)(((l-m)!)/((l+m)!)) cos(m (ϕ1-ϕ2)) (LegendreP3[l, m, cos(θ1)]) (LegendreP3[l, m, cos(θ2)])

(4*Pi*Sum[(-1)^m*SphericalHarmonicY[l, -m, θ2, ϕ2]*SphericalHarmonicY[l, m, θ1, ϕ1], 
          {m, 0,0}])/(1 + 2*l)

FunctionExpand[%]

LegendreP[l,Cos[θ1]]LegendreP[l,Cos[θ2]]+2Sum[(((l-m)!)/((l+m)!))LegendreP[l,m,Cos[θ1]]LegendreP[l,m,Cos[θ2]]Cos[m(ϕ1-ϕ2)],{m,l}]

Subtract@@@Table[{LegendreP[l,Cos[θ1]]LegendreP[l,Cos[θ2]]+2Sum[(((l-m)!)/((l+m)!))LegendreP[l,m,Cos[θ1]]LegendreP[l,m,Cos[θ2]]Cos[m(ϕ1-ϕ2)],{m,l}], LegendreP[l, Cos[θ1]*Cos[θ2] + Sin[θ1]*Sin[θ2]*Cos[ϕ1 - ϕ2]]} /. 
      {l -> RandomInteger[{0, 10}], θ1 -> RandomReal[{0, Pi}], ϕ1 -> RandomReal[{0, Pi}], 
       θ2 -> RandomReal[{0, Pi}], ϕ2 -> RandomReal[{0, Pi}]}, {100}]//Chop//Union

(Subscript[P, n])(x y-(Sqrt[1-(x^2)]) (Sqrt[1-(y^2)]) cos(α))==(Subscript[P, n])(x) (Subscript[P, n])(y)+2 (∑_(k=1)^n)(((((-1))^k) (n-k)! )/((k+n)!)) cos(k α)(Subscript[P, n]^k)(x) (Subscript[P, n]^k)(y)

LegendreP[n, x]*LegendreP[n, y] + 2*Sum[(((-1)^k*Cos[k*α]*(n - k)!)/(k + n)!)*LegendreP[n, k, x]*
     LegendreP[n, k, y], {k, 1, n}]

Subtract@@@Table[{LegendreP[n,x y-Sqrt[1-x^2] Sqrt[1-y^2] Cos[α]],LegendreP[n,x] LegendreP[n,y]+2 (∑_(k=1)^n)(((((-1))^k)Cos[k α] (n-k)!)/((k+n)!)) LegendreP[n,k,x] LegendreP[n,k,y]}/.{n->RandomInteger[{0,10}],
α->RandomReal[{-10,10},WorkingPrecision->30],x->RandomComplex[2(I+1){-1,1},WorkingPrecision->30],y->RandomComplex[2(I+1){-1,1},WorkingPrecision->30]},{100}]//Chop//Union