Laplace expansion (potential)

In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance (formula_1), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the inter-electronic repulsion.
The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors formula_2 and formula_3, then the Laplace expansion is
Here formula_2 has the spherical polar coordinates formula_6 and formula_3 has formula_8 with homogeneous polynomials of degree formula_9. Further "r"< is min("r", "r"′) and "r"> is max("r", "r"′). The function formula_10 is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics, 
Derivation.
The derivation of this expansion is simple. By the law of cosines,
We find here the generating function of the Legendre polynomials formula_13 :
Use of the spherical harmonic addition theorem 
gives the desired result.
Neumann Expansion.
A similar equation has been derived by Neumann that allows expression of formula_16 in prolate spheroidal coordinates as a series:
where formula_18 and formula_19 are associated Legendre functions of the first and second kind, respectively, defined such that they are real for formula_20. In analogy to the spherical coordinate case above, the relative sizes of the radial coordinates are important, as formula_21 and formula_22.