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 a Fourier expansion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials. We describe some of the properties of these polynomials, and use them to derive the Fourier expansion for a logarithmic fundamental solution of the polyharmonic equation. The second approach depends on the computation of parameter derivatives of Fourier series for a power-law fundamental solution of the polyharmonic equation. The resulting Fourier series is given in terms of sums over associated Legendre functions of the first kind. 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.

On a generalization of the generating function for Gegenbauer polynomials

“…Note thatx,x ∈ S d−1 , are the vectors of unit length in the direction of x, x ∈ R d respectively. For a proof of the addition theorem for hyperspherical harmonics (25), see Wen and Avery [27] and for a relevant discussion, see Section 10.2.1 in [28]. The correspondence between (24) and (25) arises from (4) and (15) namely (14).…”

“…The total number of linearly independent solutions (26) can be determined by counting the total number of terms in the sum over K in (25). Note that this formula (26) reduces to the standard result in d = 3 with a degeneracy given by 2n + 1 and in d = 4 with a degeneracy given by (n + 1) 2 .…”

“…In this paper, we only consider functions with power-law behaviour, although in future publications we will consider the logarithmic case (see [25] for the relevant Fourier expansions). Now we consider the set of hyperspherical coordinate systems which parametrize points on R d .…”

Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

“…In order to compute Fourier expansion of l d k (2.7) in separable rotationally-invariant coordinate systems, all that remains is to determine the Fourier series of g χ (see [6]). A discussion of Fourier cosine expansions for a logarithmic fundamental solution of the polyharmonic equation on R d (from g χ ) can be found in [7]. The corresponding Gegenbauer polynomial expansions for a logarithmic fundamental solution of the polyharmonic equation on R d can be found in [6].…”

Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4

“…Observe that a function u with finite volume satisfying this last expression is a solution of (4.1). Indeed, a fundamental solution of ∆ 2 is G(x) = − 1 8π |x| (see [40]). 1+|x| .…”