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Re: Poisson's integral formula doesn't work
*To*: mathgroup at smc.vnet.net
*Subject*: [mg112790] Re: Poisson's integral formula doesn't work
*From*: Mark McClure <mcmcclur at unca.edu>
*Date*: Thu, 30 Sep 2010 04:53:33 -0400 (EDT)
*References*: <201009290813.EAA27997@smc.vnet.net>
On Wed, Sep 29, 2010 at 4:13 AM, Sam Takoy <sam.takoy at yahoo.com> wrote:
> Poisson's integral formula provides a solution to the Laplace equation
> on the unit circle with Dirichlet boundary conditions.
Almost certainly a branch cut issue. You can use the periodicity of
the cosine to avoid the branch cut like so:
Integrate[Cos[phi] (1 - r^2)/(1 + r^2 - 2 r*Cos[theta - phi]),
{phi, theta - Pi, theta + Pi},
Assumptions -> {0 < r < 1, -Pi < theta < Pi}]/(2 Pi)
I wonder exactly what you want to do ultimately. It frequently makes
sense to do this kind of thing numerically. For example, you could
define u as follows:
Clear[u];
u[r_?NumericQ, theta_?NumericQ] :=
NIntegrate[Cos[phi] (1 - r^2)/(1 + r^2 - 2 r*Cos[theta - phi]),
{phi, -\[Pi], \[Pi]}]/(2 Pi)
While it works only with specific numerical values, you can treat u as
a function in just about every way. For example, you can plot its
values on a circle near the boundary to illustrate that you should
recover the cosine:
Plot[u[0.99, theta], {theta, -Pi, Pi}]
You can plot it over the disk, for that matter:
ParametricPlot3D[{r*Cos[theta], r*Sin[theta], u[r, theta]},
{r, 0, 0.99}, {theta, -Pi, Pi}]
Mark McClure
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