Exploring the Impact of Poisson Kernel on Math Functions

[matematikuvol]

Why Poisson kernel is significant in mathematics? Poisson kernel is $P_r(\theta)=\frac{1-r^2}{1-2rcos\theta+r^2}$.
http://www.math.umn.edu/~olver/pd_/gf.pdf
page 218, picture 6.15.
If we have some function for example $e^x,sinx,cosx$ what we get if we multiply that function with Poisson kernel? Thanks for the answer.

 

[lavinia]

Your question is too broad for my knowledge but the Poisson kernel in principle solves the Dirichlet problem for the unit disk in the complex plane. The Dirichlet principle says that the values of a harmonic function in a region are determined by it values on the boundary of the region. The Poisson kernel computes the function in the interior of a unit disk from its values on the boundary of the unit disk.

 

[matematikuvol]

Well ok. But for example what you get if you multiplying $P_r(\theta)\sin\theta$? Or $P^2_r(\theta)\sin\theta$? Thx for your answer. I know about that.