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

**References**:**Poisson's integral formula doesn't work***From:*Sam Takoy <sam.takoy@yahoo.com>