Poisson Kernel

In 2-D,

$$K(r,\theta) \equiv \Re\left[{R+re^{i\theta}\over R-re^{i\theta}}\right]$$

$$= \Re\left[{(R+re^{i\theta})(R-re^{-i\theta})\over (R-re^{i\theta})(R-re^{-i\theta})}\right]$$

$$= \Re\left[{R^2-rR(e^{i\theta}-e^{-i\theta})-r^2\over R^2-rR(e^{i\theta}+e^{-i\theta})+r^2}\right]$$

$$= \Re\left[{R^2+2irR\sin\theta-r^2\over R^2-2Rr\cos\theta+r^2}\right]$$

$$= {R^2-r^2\over R^2-2Rr\cos\theta+r^2}.$$ (1)

In 3-D,

$$u({\bf y})={R(R^2-a^2)\over 4\pi} \int_0^{2\pi}\int_0^\pi {f... ...\sin\theta\,d\theta\,d\phi\over(R^2+a^2-2aR\cos\gamma)^{3/2}},$$ (2)

where $a=\vert{\bf y}\vert$ and

$$\cos\gamma={\bf y}\cdot \left[{\matrix{R\cos\theta\sin\phi\cr R\sin\theta\sin\phi\cr R\cos\phi\cr}}\right].$$ (3)

The Poisson kernel for the $n$-Ball is

$$P({\bf x},{\bf z})={1\over 2-n} (D_{\bf n}{\bf v})({\bf z}),$$ (4)

where $D_{\bf n}$ is the outward normal derivative at point ${\bf z}$ on a unit $n$-Sphere and

$${\bf v}({\bf z}) = \vert{\bf z}-{\bf x}\vert^{2-n}-\vert{\bf... ...\left\vert{{\bf x}\over \vert{\bf x}\vert^2}\right\vert^{2-n}.$$ (5)

See also Poisson's Harmonic Function Formula

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1090, 1979.