The Poisson Kernel

1. Dec 27, 2011

A_B

1. The problem statement, all variables and given/known data
Can you look at Poisson's formula for a half plane as a limit case of Poisson's formula for a disk?
http://en.wikipedia.org/wiki/Poisson_kernel

I can find lots of information about the Poisson kernel for a disk, but not for the half plane. I do know on can mat the unit disk to the half plane using a Möbius transformation, but that's not what this question is asking (that comes in a later question, and is easily looked up in literature).

I can't give an "attempt at a solution" because what' I've tried hasn't produced anything useful. I tried interpreting the formula for the disk in polar coordinates, then transforming to Cartesian and looking for a suitable limit, to no avail.

This is for a project given for a first course Differential Equations.

A_B

2. Dec 30, 2011

A_B

Having worked a little more on the project, I also got to know the Poisson formulas better, and it seems to me like it doesn't make any sense at all to look at the Poisson formula for the half-plane as a limit of the Poisson formula for the disk. They are transformed into each other by a conformal map, not at all by taking some limit.
But surely the professor wouldn't put the question in a project if it can't be done?!

A_B

3. Jan 8, 2012

BVM

I'm stuck on exactly the same problem.

My initial aproach was to define the poisson kernel on a circle with radius r, and then take the limit of the poisson kernel with $r \rightarrow ∞$, but I kept on getting 0 as result.

I have found, however, that we can rewrite $$\frac{1 - r^2}{1 - 2rcos(2πt) + r^2}$$ as $$\frac{1 - z\bar{z}}{1 - z - \bar{z} + z\bar{z}}$$ with $$z =r e^{2πiθ}.$$

If that would be of any help to you...

Last edited: Jan 8, 2012
4. Jan 11, 2012

Asensegen

We seem to have pretty much the same project. Do your projects also include Schwarz–Christoffel mapping and möbiustransformation?
Did you find the solution to the limit problem yet?

I don't really see why you were taking the limit of r? My approach was rather to write the formular in terms of an arbitrary radius R and then taking the limit R -> infinity though I still don't see how this can solve the dirichlet-problem.

regards

5. Jan 11, 2012

A_B

Whaha! It's almost a convention here! I'm at KUL as well, so is BVM.
I haven't cracked it yet, but on facebook someone said she solved it starting from the Poisson kernel for the disk, but applying it to a disk of radius R with center (o, R). And then letting R go to infinity. I haven't attempted to repeat this approach yet, doing that tomorrow.

If any of you find something interesting, please let me know!

6. Jan 11, 2012

Asensegen

Did you solve the rest of it? And any idea of how thorough they expect us to be by solving those questions? There are quite a lot of terms en definitions that were not encountered in the lectures, though are necessary for solving those problems.

But yes translating the centre of the circle seems like a good approach.

good luck!

7. Jan 11, 2012

"en" oops