Suppose we have a conformal mapping from the unit disk in the $\omega$ plane onto the exterior of a polygon in the $z$ plane.

The Schwarz-Christoffel mapping in this case is defined as: $$f(u) = A - C \int^u w^{-2}\prod_{k=1}^{n} (1-\frac{\omega}{u_k})^{1-\alpha_k} d\omega.$$

Meaning that we determine the vertex $z_k$ with the mapping $f(u_k)$ from the respective prevertex.

I have provided an image from Discroll and Trefethens' book on the subject (PDF sample with the image is freely available).

enter image description here

I would like to compute a factor from an integral over function analytic inside a polygon in the complex plane.

For this I will use Gauss-Kronrod quadrature, specifically MATLAB's quadgk command with 'waypoints'

I ll just pick my function arbitrarily as $g = \exp(z)$.

What I want to compute is:


To use quadgk I have to define a function handle for $g$. But I am confused on how to do that with the SC-Mapping f(u). I know I can extract the prevertices and interior angles $\alpha_k$ with the SC-toolbox for any polygon like so:

PSI = extermap(drawpoly)
p = polygon(PSI);
a = parameters(PSI);
  1. How do I compute the integral $\int^u w^{-2}\prod_{k=1}^{n} (1-\frac{\omega}{u_k})^{1-\alpha_k} d\omega$ ? It's a polygonal integral in the complex plane so should I use quadgk as well ?
  2. How do I set the complex constants $A$ and $C$ ?

    b = a.constant

seems to refer to the logarithmic capacity of the polygonal domain.


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