Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

Abstract

In even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution for the polyharmonic equation has logarithmic behavior. We give two approaches for developing an azimuthal Fourier expan- sion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials which we call logarithmic polynomials. The second approach depends on the computation of parameter derivatives of Fourier series expressions for a power-law fundamental solution of the polyharmonic equation. We conclude by comparing the two approaches and giving the azimuthal Fourier series for a logarithmic fundamental solution of the polyharmonic equation in rotationally-invariant coordinate systems. Solutions of the polyharmonic equation (powers of the Laplacian operator) are ubiquitous in many areas of computational, pure, applied mathematics, physics and en- gineering. We concern ourselves, in this paper, with a fundamentalsolution of the poly- harmonic equation (Laplace, biharmonic, etc.), which by convolution yields a solution to the inhomogeneous polyharmonic equation. Solutions to inhomogeneous polyhar- monic equations are useful in many physical applications including those areas related to Poisson's equation such as Newtonian gravity, electrostatics, magnetostatics, quan- tum direct and exchange interactions (cf. Section 1 in (3)), etc. Furthermore, applica- tions of higher-powers of the Laplacian include such varied areas as minimal surfaces (12), Continuum Mechanics (8), Mesh deformation (6), Elasticity (9), Stokes Flow (7), Geometric Design (20), Cubature formulae (17), mean value theorems (cf. Pizzetti's formula) (13), and Hartree-Fock calculations of nuclei (21). Closed-form expressions for the Fourier expansions of a logarithmic fundamental solution for the polyharmonic equation are extremely useful when solving inhomoge- neous polyharmonic problems in even-dimensionalEuclidean space, especially when a degreeof rotationalsymmetry is involved. A fundamentalsolution of the polyharmonic equation on d-dimensionalEuclidean space R d has two argumentsand therefore maps from a 2d-dimensional space to the reals. Solutions to the inhomogeneous polyhar- monic equation can be obtained by convolution of a fundamental solution with an in- tegrable source distribution. Eigenfunction decompositions of a fundamental solution