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Poisson's integral formula doesn't work

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112746] Poisson's integral formula doesn't work
  • From: Sam Takoy <sam.takoy at yahoo.com>
  • Date: Wed, 29 Sep 2010 04:13:20 -0400 (EDT)

Hi,

Poisson's integral formula provides a solution to the Laplace equation 
on the unit circle with Dirichlet boundary conditions. The formula can 
be found here in equation (8):

http://mechse.illinois.edu/research/dstn/teaching_files2/poissonformula.pdf


Here's my implementation (I hope it's correct)

PoissonFormula[h_] :=
   1/(2 Pi)
      Integrate[(1 - #1^2)/(1 + #1^2 - 2 #1 Cos[#2 - alpha])
        h[alpha], {alpha, 0, 2 Pi}] &;
$Assumptions = r > 0 && r < 1 && \[Alpha] > 0 && \[Alpha] < 2 Pi;
PoissonFormula[Cos][r, \[Alpha]]

The result should be

r Cos[\[Alpha]]

Instead, I get

((-1 + r^2) Cos[\[Alpha]])/(2 r)

What's the problem here?

Thanks!

Sam


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