[Paper Review] Spherical Harmonics in p Dimensions

Christopher Frye, Costas Efthimiou|arXiv (Cornell University)|May 16, 2012

Electromagnetic Scattering and Analysis5 citations

TL;DR

This paper provides a comprehensive, accessible derivation of spherical harmonics and associated Legendre polynomials in p-dimensional Euclidean space, using orthogonal polynomials and harmonic analysis. It establishes the key generating function identity $\sum_{n=0}^{\infty} r^n N(p,n) P_n(t) = \frac{1 - r^2}{(1 - 2rt + r^2)^{p/2}}$, which generalizes the Poisson kernel and enables solution of Laplace's equation via separation of variables and Green's function methods in arbitrary dimensions.

ABSTRACT

The authors prepared this booklet in order to make several useful topics from the theory of special functions, in particular the spherical harmonics and Legendre polynomials for any dimension, available to undergraduates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before beginning the main subject matter. The reader is assumed to have knowledge of multivariable calculus and linear algebra as well as some level of comfort with reading proofs.

Motivation & Objective

• To develop a self-contained, accessible treatment of spherical harmonics in $\mathbb{R}^p$ for advanced undergraduates in physics and mathematics.
• To generalize the theory of spherical harmonics and Legendre polynomials from $\mathbb{R}^3$ to arbitrary dimensions $p$.
• To establish the orthogonality, completeness, and generating function structure of spherical harmonics in $p$ dimensions.
• To apply the theory to solve boundary value problems for Laplace's equation using the method of images and Green's functions.
• To derive the Poisson kernel identity $\sum_{n=0}^{\infty} r^n N(p,n) P_n(t) = \frac{1 - r^2}{(1 - 2rt + r^2)^{p/2}}$ as a central result.

Proposed method

• Derives the Laplacian in $p$-dimensional spherical coordinates using the chain rule and coordinate transformations.
• Introduces harmonic homogeneous polynomials and defines spherical harmonics as their restrictions to the unit sphere $S^{p-1}$.
• Establishes orthogonality and completeness of spherical harmonics using Hilbert space theory and inner product structure.
• Applies Rodrigues' formula and recurrence relations to construct orthogonal polynomials in $p$ dimensions.
• Constructs the Green's function for the Laplacian in $\mathbb{R}^p$ using the method of images, placing a source and its image charge outside the unit ball.
• Derives the Poisson kernel representation for harmonic functions via Green's theorem and comparison of integral solutions.

Experimental results

Research questions

• RQ1 How can spherical harmonics be generalized from $\mathbb{R}^3$ to arbitrary dimensions $p$ using harmonic polynomials and orthogonal function theory?
• RQ2 What is the explicit form of the generating function for Legendre polynomials in $p$ dimensions, and how does it relate to the Poisson kernel?
• RQ3 How can the method of images be applied to construct the Green's function for the Laplacian in $\mathbb{R}^p$?
• RQ4 What is the integral representation of harmonic functions on the unit ball in $\mathbb{R}^p$ in terms of boundary data?
• RQ5 How does the completeness of spherical harmonics in $L^2(S^{p-1})$ ensure the expansion of harmonic functions in $p$ dimensions?

Key findings

• The paper derives the generating function identity $\sum_{n=0}^{\infty} r^n N(p,n) P_n(t) = \frac{1 - r^2}{(1 - 2rt + r^2)^{p/2}}$, which generalizes the Poisson kernel to $p$ dimensions.
• The spherical harmonics $Y_{n,j}$ form a complete orthonormal basis for $L^2(S^{p-1})$, enabling expansion of any square-integrable function on the sphere.
• The Green's function for the Laplacian in $\mathbb{R}^p$ is constructed as $G(x;x_0) = \frac{1}{(2-p)\Omega_{p-1}} \left( \frac{1}{|x - x_0|^{p-2}} - \frac{1}{(|x_0| \cdot |x - x_0'|)^{p-2}} \right)$, ensuring vanishing on the unit sphere.
• The solution to the Dirichlet problem on the unit ball in $\mathbb{R}^p$ is given by $V(x_0) = \frac{1}{\Omega_{p-1}} \int_{S^{p-1}} f(\xi) \frac{1 - |x_0|^2}{(1 + |x_0|^2 - 2|x_0|\cos\theta)^{p/2}} d\Omega_{p-1}$, matching the Poisson kernel form.
• The dimension-dependent normalization factor $N(p,n)$, counting the number of linearly independent spherical harmonics of degree $n$ in $p$ dimensions, is explicitly used in the generating function.
• The method of images is rigorously applied in $p$ dimensions to construct the Green's function, with the image point located at $x_0' = x_0 / |x_0|^2$.