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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 *r*^{n} of the distance *r* between two points (*r* _{1}, ϑ_{1}, φ_{1}) and (*r* _{2}, ϑ_{2}, φ_{2}) is expanded in terms of Legendre polynomials of cos ϑ_{12}. The coefficients are homogeneous functions of *r* _{1} and *r* _{2} 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 *r* _{1} tends to *r* _{2} and of the analytic continuation of the functions past these singularities. Expressions symmetric in *r* _{1} and *r* _{2} 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*(*r* _{1} + *r* _{2}) or *f*[(*r* _{1} ^{2} + *r* _{2} ^{2})^{½}].

© 1964 The American Institute of Physics

Received 16 August 1963
Published online 22 December 2004