Question 8.21: Find a transformation that maps the unit circle in the ζ pla...
Find a transformation that maps the unit circle in the \zeta plane onto a polygon in the w plane.
The x axis in the z plane can be mapped onto a polygon of the w plane by the Schwarz-Christoffel transformation
and the upper half of the z plane maps onto the interior of the polygon.
A transformation that maps the upper half of the z plane onto the unit circle in the \zeta plane is
\zeta=\frac{i-z}{i+z} (2)
on replacing w by \zeta and taking \theta=\pi, z_{0}=i in equation (8.8), page 246.
w=e^{i\theta_{0}}\left({\frac{z-z_{0}}{z-{\bar{z}}_{0}}}\right) (8.8)
Hence, z=i\{(1-\zeta) /(1+\zeta)\} maps the unit circle in the \zeta plane onto the upper half of the z plane.
If we let x_{1}, x_{2}, \ldots, x_{n} map into \zeta_{1}, \zeta_{2}, \ldots, \zeta_{n}, respectively, on the unit circle, then we have for k=1,2, \ldots, n.
z-x_{k}=i\left(\frac{1-\zeta}{1+\zeta}\right)-i\left(\frac{1-\zeta_{k}}{1+\zeta_{k}}\right)=\frac{-2 i\left(\zeta-\zeta_{k}\right)}{(1+\zeta)\left(1+\zeta_{k}\right)}
Also, d z=-2 i d \zeta /(1+\zeta)^{2}. Substituting into (1) and simplifying using the fact that the sum of the exponents \left(\alpha_{1} / \pi\right)-1,\left(\alpha_{2} / \pi\right)-1, \ldots,\left(\alpha_{n} / \pi\right)-1 is -2 , we find the required transformation
where A^{\prime} is a new arbitrary constant.