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Spherical Harmonic Addition Theorem

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A formula also known as the Legendre addition theorem which is derived by finding Green's functions for the spherical harmonic expansion and equating them to the generating function for Legendre polynomials. When gamma is defined by

cosgamma=costheta_1costheta_2+sintheta_1sintheta_2cos(phi_1-phi_2),
(1)

The Legendre polynomial of argument gamma is given by

P_l(cosgamma)=(4pi)/(2l+1)sum_(m=-l)^(l)(-1)^mY_l^m(theta_1,phi_1)Y_l^(-m)(theta_2,phi_2)
(2)
=(4pi)/(2l+1)sum_(m=-l)^(l)Y_l^m(theta_1,phi_1)Y^__l^m(theta_2,phi_2)
(3)
=P_l(costheta_1)P_l(costheta_2)+2sum_(m=1)^(l)((l-m)!)/((l+m)!)P_l^m(costheta_1)P_l^m(costheta_2)cos[m(phi_1-phi_2)].
(4)

Another version of the formula can be given as

P_n(xy-sqrt(1-x^2)sqrt(1-y^2)cosalpha) =P_n(x)P_n(y)+2sum_(k=1)^n((-1)^k(n-k)!)/((k+n)!)cos(kalpha)P_n^k(x)P_n^k(y)
(5)

(O. Marichev, pers. comm., Jan. 15, 2008).

See also

Associated Legendre Polynomial, Legendre Polynomial, Spherical Harmonic

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References

Arfken, G. "The Addition Theorem for Spherical Harmonics." §12.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 693-695, 1985.

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Spherical Harmonic Addition Theorem

Cite this as:

Weisstein, Eric W. "Spherical Harmonic Addition Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalHarmonicAdditionTheorem.html

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