Solving Laplace’s equation in the upper half plane

In the previous post, I said that solving Laplace’s equation on the unit disk was important because the unit disk is a sort of “hub” of conformal maps: there are references and algorithms for mapping regions to and from a disk conformally.

The upper half plane is a sort of secondary hub. You may want to map two regions to and from each other via a half plane. And as with the disk, there’s an explicit solution to Laplace’s equation on a half plane.

Another reason to be interested in Laplace’s equation on a half plane is the connection to the Hilbert transform and harmonic conjugates.

Given a continuous real-valued function u on the real line, u can be extended to a harmonic function on the upper half plane by taking the convolution of u with the Poisson kernel, a variation on the Poisson kernel from the previous post. That is, for y > 0,

u(x + iy) = \frac{1}{\pi} \int_{-\infty}^\infty \frac{y}{(x-t)^2 + y^2}\, u(t)\, dt

This gives a solution to Laplace’s equation on the upper half plane with boundary values given by u on the real line. The function u is smooth on the upper half plane, and its limiting values as y → 0 is continuous.

Furthermore, u is the real part of an analytic function f = u + iv. The function v is the harmonic conjugate of u, and also equals the Hilbert transform of u.

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