Poisson's Harmonic Function Formula

Let $\phi(z)$ be a Harmonic Function. Then

$$\phi(z_0)={1\over 2\pi} \int_0^{2\pi} K(r,\theta)\phi(z_0+re^{i\theta})\,d\theta,$$ (1)

where $R=\vert z_0\vert$ and $K(r,\theta)$ is the Poisson Kernel. For a Circle,

$$u(x,y)={1\over 2\pi} \int_0^{2\pi} u(a\cos\phi,a\sin\phi){a^2-R^2\over a^2+R^2-2ar\cos(\theta-\phi)} \,d\phi.$$ (2)

For a Sphere,

$$u(x,y,z) = {1\over 4\pi a}\int\!\!\!\int _S u {a^2-R^2\over (a^2+R^2-2aR\cos\theta)^{3/2}}\,dS,$$ (3)

where

$$\cos\theta\equiv {\bf x}\cdot {\boldsymbol{\xi}}.$$ (4)

See also Circle, Harmonic Function, Poisson Kernel, Sphere

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 373-374, 1953.