Poisson kernel
A function P_r(\psi) that reproduces the disk-series solution of Laplace's equation as a single integral against the boundary data, instead of an infinite sum.
For the Dirichlet problem on the unit disk, the Poisson kernel P_r(\psi)=\dfrac{1-r^2}{1-2r\cos\psi+r^2} is the function such that u(r,\theta)=\dfrac1{2\pi}\displaystyle\int_{-\pi}^{\pi}f(\varphi)\,P_r(\theta-\varphi)\,d\varphi solves \nabla^2u=0 with u(1,\theta)=f(\theta) -- the same solution the disk's Fourier series produces, summed in closed form.
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