aspcol.sphericalharmonics.gaunt_coefficient

aspcol.sphericalharmonics.gaunt_coefficient(l1, m1, l2, m2, l3)

Gaunt coefficient G(l1, m1, l2, m2, l3)

Defined on page 83, equation (3.71) in [martinMultiple2006]. Argument order is the same as in the reference

Parameters:

  • l1 (int) – Spherical harmonic order

  • m1 (int) – Spherical harmonic degree

  • l2 (int) – Spherical harmonic order

  • m2 (int) – Spherical harmonic degree

  • l3 (int) – Spherical harmonic order

Returns:

G – Gaunt coefficient for the given arguments

Return type:

float

References

[martinMultiple2006] P. A. Martin, Multiple scattering: Interaction of time-harmonic waves with N obstacles, vol. 107. in Encyclopedia of mathematics and its applications, vol. 107. Cambridge, UK: Cambridge University Press, 2006.

Notes

The relationship between this and triple_harmonic_integral (the latter is the definition of gaunt coefficient given by Sympy) is I(l1, l2, l3, m1, m2, m3) = delta(m1+m2+m3,0) * (-1)**(m3) * gaunt(l1, m1, l2, m2, l3)

Can be seen on the final equation on page 328 in Multiple scattering: Interaction of time-harmonic waves with N obstacles by P. A. Martin, 2006.

A recursive algorithm can be found in: Fast evaluation of Gaunt coefficients: recursive approach - Yu-lin Xu, 1997