- Poisson kernel
In
potential theory , the Poisson kernel is thederivative of theGreen's function for the two-dimensionalLaplace equation , undercircular symmetry , usingDirichlet boundary condition s. It is used for solving the two-dimensionalDirichlet 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 theupper 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 theDirichlet 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 D = {z:|z|<1} 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 0-387-95218-7External links
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