Poisson Kernel: Examining Half Plane Limit Case

[A_B]

Homework Statement

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.

Thanks in advance
A_B

 

[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

 

[BVM]

I'm stuck on exactly the same problem.

My initial approach 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...

 

[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 formula 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

 

[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!

 

[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!

 

[Asensegen]

"en" oops