the Poisson integral formula
/ PWAH-sohn /
For Cauchy's integral formula, the boundary values of a holomorphic function on a circle reconstruct it inside. The Poisson integral formula is the real-variable twin of that idea for harmonic functions: give the values of a harmonic function on the boundary circle of a disk, and the formula reconstructs its value at every interior point as a weighted average of the boundary data. It is the explicit solution of the Dirichlet problem on a disk.
On the unit disk, suppose we want the harmonic u with prescribed continuous boundary values u(e^(i t)) = g(t) on the unit circle. Then for an interior point z = r e^(i theta) with 0 <= r < 1, Poisson's formula says u(r e^(i theta)) = (1 / (2 pi)) times the integral from 0 to 2 pi of P_r(theta - t) g(t) dt, where P_r(phi) = (1 - r^2) / (1 - 2 r cos phi + r^2) is the Poisson kernel. Read it as an averaging: each boundary value g(t) is weighted by P_r(theta - t), which is large when t is near the angle theta (close to z) and small when t is far, so points of the boundary nearest z count most. The weights are nonnegative and integrate to 2 pi, making the formula a genuine weighted average — that is why the maximum principle drops out for free. As r -> 0 the kernel flattens and you recover the mean-value property u(0) = average of g.
This single formula does a great deal: it proves the Dirichlet problem on a disk is always solvable for continuous boundary data, it shows the solution is smooth (indeed real-analytic) inside no matter how rough g is, and it is the seed for solving the problem on any nicely-bounded region by conformal transplantation. A practical caution: the formula recovers u only inside the open disk; the limit as z approaches the boundary equals g(t) at every point where g is continuous, but if g has a jump, the interior values smoothly average across the jump rather than reproducing it.
Take boundary data g(t) = cos t. The Poisson integral returns u(r e^(i theta)) = r cos theta = x, the unique harmonic function on the disk equal to cos t on the boundary — matching the Dirichlet-problem answer found by inspection.
Poisson's formula turns boundary data cos t into the harmonic interior r cos theta.
The formula as stated is for the unit disk; on a disk of radius R centered at z_0 replace r by |z - z_0|/R appropriately. For regions other than disks you first map conformally to a disk — there is no equally simple universal kernel for an arbitrary region.