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Poisson Kernel

In 2-D,

$\displaystyle K(r,\theta)$ $\textstyle \equiv$ $\displaystyle \Re\left[{R+re^{i\theta}\over R-re^{i\theta}}\right]$  
  $\textstyle =$ $\displaystyle \Re\left[{(R+re^{i\theta})(R-re^{-i\theta})\over (R-re^{i\theta})(R-re^{-i\theta})}\right]$  
  $\textstyle =$ $\displaystyle \Re\left[{R^2-rR(e^{i\theta}-e^{-i\theta})-r^2\over R^2-rR(e^{i\theta}+e^{-i\theta})+r^2}\right]$  
  $\textstyle =$ $\displaystyle \Re\left[{R^2+2irR\sin\theta-r^2\over R^2-2Rr\cos\theta+r^2}\right]$  
  $\textstyle =$ $\displaystyle {R^2-r^2\over R^2-2Rr\cos\theta+r^2}.$ (1)

In 3-D,

u({\bf y})={R(R^2-a^2)\over 4\pi} \int_0^{2\pi}\int_0^\pi {f...
\end{displaymath} (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].
\end{displaymath} (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}),
\end{displaymath} (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}.
\end{displaymath} (5)

See also Poisson's Harmonic Function Formula


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

© 1996-9 Eric W. Weisstein