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,

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*.