Can anyone tell me how to calculate the Poisson kernel for the upper half plane? I am able to calculate it for the unit disc and I know the unit disc and the upper half plane are conformally equivalent, do I need this?
[Math] Poisson kernel for upper half plane
complex-analysis
Best Answer
This is a good exercise. Let $\phi$ be the conformal mapping of the half plane to the unit disk.
To create a harmonic function on $\mathbb{H}$ which agrees with $f$ on the real line, one good strategy would be to translate it to the unit disk. Using the Poisson kernel for the disk, we can find a harmonic function on the disk which agrees with $f\circ \phi^{-1}$ on the boundary. Compose it with $\phi$ (which is also harmonic) to get a function which is harmonic on $\mathbb{H}$ that agrees with $f$ on the real line.
This is an outline, in the sense that to derive the Poisson kernel for the upper half plane, you have to power through some algebraic manipulations. That is messy, but not hard (especially if you know the answer).