11.4 Spherical harmonics and Legendre polynomials
For Laplace’s equation in spherical coordinates, separate \(u = R(r)Y(\theta , \varphi )\): \(r^2R'' + 2rR' = \lambda R\) and \(\Delta _SY = -\lambda Y\). The angular problem has solutions regular on the whole sphere only for \(\lambda = l(l + 1)\), \(l = 0, 1, 2, \ldots \), and then has \(2l + 1\) independent solutions, the spherical harmonics \(Y_l^m(\theta , \varphi ) = P_l^{|m|}(\cos \theta )e^{im\varphi }\), \(|m| \leq l\), with \(P_l^m\) the associated Legendre functions. The radial equation has the solutions \(r^l\) and \(r^{-l - 1}\). So the harmonic functions in a ball are sums of \(r^lY_l^m\), and those outside, decaying at infinity, are sums of \(r^{-l - 1}Y_l^m\): the multipole expansion. The spherical harmonics are orthogonal on the sphere and complete, and they describe the gravitational field of the Earth, the angular part of atomic orbitals, and the fluctuations of the cosmic microwave background.
When the data do not depend on \(\varphi \), only \(m = 0\) occurs, and \(P_l^0 = P_l\) is the Legendre polynomial of degree \(l\), with \(\int _{-1}^1P_kP_l\,d\mu = \frac {2}{2l + 1}\delta _{kl}\). The solution of the Dirichlet problem in the unit ball with \(u = f(\theta )\) on the sphere is \[ u(r, \theta ) = \sum _{l = 0}^\infty A_lr^lP_l(\cos \theta ), \qquad A_l = \frac {2l + 1}2\int _{-1}^1f\,P_l(\mu )\,d\mu \quad (\mu = \cos \theta ) . \]
Example 11.4.1. Let the upper hemisphere be held at potential \(1\) and the lower at \(-1\). Then \(f\) is odd in \(\mu \), so only odd \(l\) occur, and \(A_l = (2l + 1)\int _0^1P_l\,d\mu \): \[ u = \tfrac 32rP_1(\cos \theta ) - \tfrac 78r^3P_3(\cos \theta ) + \tfrac {11}{16}r^5P_5(\cos \theta ) - \cdots . \] On the axis, \(\theta = 0\), the Poisson kernel of the ball (Section 9) gives the closed form \(u = \bigl (1 - (1 - z^2)/\sqrt {1 + z^2}\bigr )/z\) at height \(z\), against which the series can be checked (Software 11.6.3).
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