Poisson integral formula

Poisson integral formula

In mathematics, the Poisson integral formula gives an explicit solution to the Dirichlet problem for Laplace's equation in a ball in Euclidean space Rn.

If "u" is a harmonic function in the ball in Rn centered at the origin with radius "R", then the formula states

: u(x) = frac{R^2 - |x|^2}{omega_n R} intlimits_{partial B_R} frac{u(y)}{|x - y|^n},dS(y)

where omega_n is the surface area of the unit sphere. The integration is performed over the surface of the ball, with unit surface area dS(y).

References

*D. Gilbarg, N. Trudinger "Elliptic Partial Differential Equations of Second Order". ISBN 3-540-41160-7.

See also

* Schwarz formula

External links

*
*
* [http://math.fullerton.edu/mathews/c2003/PoissonIntegralMod.html Poisson Integral Module by John H. Mathews]



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