Generalization of Laplace's Expansion to Arbitrary Powers and Functions of the Distance between Two Points

Abstract

In analogy to Laplace's expansion, an arbitrary power rn of the distance r between two points (r1, ϑ1, φ1) and (r2, ϑ2, φ2) is expanded in terms of Legendre polynomials of cos ϑ12. The coefficients are homogeneous functions of r1 and r2 of degree n satisfying simple differential equations; they are solved in terms of Gauss' hypergeometric functions of the variable (r</r>)2. The transformation theory of hypergeometric functions is applied to describe the nature of the singularities as r1 tends to r2 and of the analytic continuation of the functions past these singularities. Expressions symmetric in r1 and r2 are obtained by quadratic transformations; for n = −1 and n = −2; one of these has previously been given by Fontana. Some three-term recurrence relations between the radial functions are established, and the expressions for the logarithm and the inverse square of the distance are discussed in detail. For arbitrary analytic functions f(r), three analogous expansions are derived; the radial dependence involves spherical Bessel functions of (r<∂/∂r>) of of related operators acting on f(r>), f(r1 + r2) or f[(r12 + r22)½].