Complex Analysis: Examples & Questions Solved with Poisson's & Cauchy's Formulas

[esisk]

Hope this does not sound vague!

1) I a looking at the Poisson's formula for the disk. Can somebody give me an example how one uses this, or a question where we use it to solve the problem. What is it exactly saying that Cauchy's formula is not saying? Thank you

2) Can somebody give me an example wherby I use the inversion with respect to a circle (unit circle or otherwise) and the problem becomes easier. I guess I am asking: how do I make use of this notion.
Thank you

 

[hamster143]

Cauchy's formula gives you values inside the disk if you know complex values on the boundary. Poisson's formula allows you to reconstruct the function from just the real component. As such, it has some use outside complex analysis, because it allows you to solve the two dimensional Laplace equation.