We show that an exact expression for the Greens function in cylindrical coordinates is

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace's equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace's equation on the R-radius hyperboloid. For d ≥ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We develop an alternative based on an old, but hitherto forgotten, expression for the inverse distance between two points that builds on azimuthal symmetry. This alternative should have wide applicability throughout physics and astronomy, both for computation and for the insights it provides through its emphasis on different symmetries and structures than are familiar from the standard treatment. We compare and contrast the two methods, develop addition theorems for Legendre functions of the second kind and a number of useful analytical expressions for these functions. Two-electron “direct” and “exchange” integrals in many-electron quantum systems are evaluated to illustrate the procedure, which is more compact than the standard one using Wigner coefficients and Slater integrals.

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of several previously derived formulae such as Heine's formula and Heine's reciprocal square-root identity. We also show how this expansion can be used to compute hyperspherical harmonic expansions for power-law fundamental solutions of the polyharmonic equation.