the Poisson kernel
/ PWAH-sohn /
When the Poisson formula averages boundary values to build a harmonic function inside the disk, something has to decide how much weight each boundary point gets. That weighting function is the Poisson kernel. Think of it as a spotlight aimed from the interior point out at the boundary circle: it shines brightest on the part of the rim nearest the point and dims for far-away parts, so nearby boundary data counts most.
On the unit disk the Poisson kernel is P_r(phi) = (1 - r^2) / (1 - 2 r cos phi + r^2), where r in [0, 1) is the distance of the interior point from the center and phi is the angular gap between the interior point's direction and the boundary point. It has three properties that make the averaging work: it is nonnegative everywhere; for each fixed r it integrates to 2 pi over phi from 0 to 2 pi (so dividing by 2 pi makes the weights sum to one); and as r -> 1 it concentrates into a sharp spike at phi = 0, becoming an approximate identity that pulls interior values toward the matching boundary value. There is also a tidy complex form: P_r(phi) is the real part of (1 + z) / (1 - z) evaluated with z = r e^(i phi), tying the kernel directly to a holomorphic function.
The Poisson kernel is the engine inside the Poisson integral formula, and it is the reason that formula simultaneously solves the Dirichlet problem and respects the maximum principle (a nonnegative, mass-one weighting can never produce a value outside the range of the boundary data). It is also the harmonic analogue of the Cauchy kernel 1 / (z - z_0): both reconstruct an interior value from boundary data, one for harmonic functions, the other for holomorphic ones. The caveat to remember: the spike at r near 1 is what makes boundary values get reproduced, but it also means that for rough boundary data the convergence to the boundary is in an averaged sense, not pointwise at every single bad point.
At the center r = 0, P_0(phi) = (1 - 0) / (1 - 0 + 0) = 1, a flat weighting; so the Poisson formula at the center is just the plain average u(0) = (1 / (2 pi)) times the integral of g(t) dt — the mean-value property.
At the center the kernel is constant, so Poisson's average collapses to the mean-value property.
P_r(phi) is positive precisely because 1 - 2 r cos phi + r^2 = |1 - r e^(i phi)|^2 > 0 for r < 1; this positivity is not cosmetic — it is exactly what guarantees the resulting function honors the maximum principle.