This question arises out of confusion about some aspects of the Poisson integral in $R^2$, simplified here to a function f on the boundary of the unit circle $ \partial C$ and on C,
$u(r,\phi) = \frac{1}{2\pi} \int_0^{2\pi} \frac{ f (\theta) (1-r^2) d\theta}{1-2r \cos(\theta-\phi)+r^2}$
The intuition behind the derivation of the integral is described nicely here and based on this sort of intuition I wondered whether there might be an approach that did not involve the use of "inversive" geometry.
The article cited mentions that we would typically be considering harmonic functions (temperature, e.g.) and that given an arbitrary piecewise continuous assignment of values to the boundary $\partial C$, "there always exists a harmonic function in R that takes on these values as the boundary is approached."
Well, suppose $f(\phi) = \phi^2 $? There is a discontinuity at $(r = 1, \phi = 0, 2 \pi )$. According to the article, if we are approaching $ \mathit{continuous }$ boundary points this is not an issue.
So for the question(s). If I guess--naively-- that the influence of a point on the boundary on a point inside the circle is inversely proportional to the square of the distance between the two points, I might approximate the integral as
$ u \approx \frac{\sum_{i=1}^n \frac{f(\phi_i) }{(d_i)^2}}{\sum_{i=1}^n \frac{1}{(d_i)^2}}$
This approximation is not great, but it yields reasonable values near the center of the circle.
Is there a straightforward Riemann-sum approximation of the integral along these lines? Or are we stuck with the geometry of the article? Is the partial agreement I get for my guess fortuitous?
We could assign pretty outrageous piecewise-continuous values to the boundary--do every one of these correspond to a theoretically possible (for example) heat distribution?
Thanks for any insights, edits.