Poisson kernel

In potential theory, the Poisson kernel is the derivative of the Green's function for the two-dimensional Laplace equation, under circular symmetry, using Dirichlet boundary conditions. It is used for solving the two-dimensional Dirichlet problem.

In practice, there are many different forms of the Poisson kernel in use. For example, in complex analysis, the Poisson kernel for a disc is often used as is the Poisson kernel for the upper half-plane, and both of these can be extended into n-dimensional space.

In the complex plane, the Poisson kernel for the unit disc is given by

:$P\_r(\; heta)\; =\; sum\_\{n=-infty\}^infty\; r^e^\{in\; heta\}\; =\; frac\{1-r^2\}\{1-2rcos\; heta\; +r^2\}\; =\; Releft(frac\{1+re^\{i\; heta\{1-re^\{i\; heta\; ight).$

This can be thought of in two ways: either as a function of $r$ and $heta$, or as a family of functions of $heta$ indexed by $r$.

One of the main reasons for the importance of the Poisson kernel in complex analysis is that the Poisson integral of the Poisson kernel gives a solution of the Dirichlet problem for the disc. The Dirichlet problem asks for a solution to Laplace's equation on the unit disk, subject to the Dirichlet boundary condition. If is the unit disc in C, and if "f" is a continuous function from $partial\; D$ into R, then the function "u" given by

:$u(re^\{i\; heta\})\; =\; frac\{1\}\{2pi\}int\_\{-pi\}^pi\; P\_r(\; heta-t)f(e^\{it\})dt$

is harmonic in D and agrees with "f" on the boundary of the disc.

For the ball of radius r, $B\_\{r\}$, in Rⁿ, the Poisson kernel takes the form

:$P(x,zeta)\; =\; frac\{r^2-|x|^2\}\{romega\; \_\{n\}|x-zeta|^n\}$

where $xin\; B\_\{r\}$, $zetain\; S$ (the surface of $B\_\{r\}$), and $omega\; \_\{n\}$ is the surface area of the unit ball.

Then, if "u"("x") is a continuous function defined on "S", the corresponding result is that the function "P" ["u"] ("x") defined by

:$P\; [u]\; (x)\; =\; int\_S\; u(zeta)P(x,zeta)dsigma(zeta)$

is harmonic on the ball $B\_\{r\}$.

References

*cite book | author = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | id=ISBN 0-387-90328-3
*cite book | author = S. Axler, P. Bourdon, W. Ramey | title = Harmonic Function Theory | publisher = Springer-Verlag | year = 1992 | id=ISBN 0-387-95218-7

External links

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