Poissonintegral

PoissonIntegral refers to the Poisson integral operator used in potential theory and complex analysis to recover harmonic functions from boundary data on a circle or a line. It is named after Siméon Denis Poisson and plays a central role in solving Dirichlet problems for Laplace’s equation in standard domains.

In the unit disk, suppose f is defined on the unit circle. The Poisson kernel is P_r(φ) = (1 − r^2)/(1 − 2r cos φ + r^2) for 0 ≤ r < 1. The Poisson integral of f is

u(r, θ) = (1/2π) ∫_0^{2π} P_r(θ − φ) f(φ) dφ.

The resulting function u is harmonic in the disk. As r → 1−, u(r, ·) converges to f in the appropriate sense: it tends to f uniformly if f is continuous, and converges in L^p for suitable p. This provides a solution to the Dirichlet problem with boundary data f.

In the upper half-plane, the Poisson kernel is P_y(x) = (1/π) y / (x^2 + y^2). For boundary data f on the real line, the Poisson integral is

u(x, y) = ∫_{−∞}^{∞} f(t) P_y(x − t) dt, for y > 0.

The function u is harmonic in the upper half-plane and tends to f on the boundary in the appropriate sense; for many f this convergence holds almost everywhere as y → 0+.

Generalizations exist to higher dimensions via Poisson kernels for the unit ball and the upper half-space. The PoissonIntegral thus provides a canonical, linear method to extend boundary data to harmonic functions.