Question 8.26: Prove that the Schwarz –Christoffel transformation of Proble...
Prove that the Schwarz-Christoffel transformation of Problem 8.17 maps the upper half plane onto the interior of the polygon.
It suffices to prove that the transformation maps the unit circle onto the interior of the polygon, since we already know [Problem 8.11] that the upper half plane can be mapped onto the unit circle.
Suppose that the function mapping the unit circle C in the z plane onto polygon P in the w plane is given by w=f(z) where f(z) is analytic inside C.
We must now show that to each point a inside P, there corresponds one and only one point, say z_{0}, such that f\left(z_{0}\right)=a.
Now, by Cauchy’s integral formula, since a is inside P,
\frac{1}{2 \pi i} \oint\limits_{P} \frac{d w}{w-a}=1
Then, since w-a=f(z)-a,
\frac{1}{2 \pi i} \oint\limits_{C} \frac{f^{\prime}(z)}{f(z)-a} d z=1
But f(z)-a is analytic inside C. Hence, from Problem 5.17, we have shown that there is only one zero (say z_{0} ) of f(z)-a inside C, i.e., f\left(z_{0}\right)=a, as required.