Prove. Consider a polygon in the w plane having vertices at w1, w2, ... , wn with corresponding interior angles α1, α2, ... , αn respectively. See Fig. 1b.
A transformation which maps the upper half plane R of the z plane onto the interior R' of the polygon and the real axis of the z plane onto the boundary of the polygon is given by
where the points x1, x2, ... , xn on the real axis of the z plane are mapped respectively into the points w1, w2, ... , wn of the polygon and A and B are complex constants.
Proof. The following will be more demonstration than proof. It will give intuitive insight into the transformation.
First observe that from 1] we obtain
3] arg dw = arg dz + arg A + (α1/π -1) arg(z - x1) + (α2/π -1) arg (z - x2) + ... + (αn/π -1) arg(z - xn)
We direct our attention to what occurs as z moves along the real axis from the left, tracing out the polygon as it moves. As z passes each of the zeros at x1, x2, ... , xn an event occurs. There is an abrupt change in argument at that point. And this abrupt change in argument causes an abrupt change arg dw which causes w to make an abrupt change in direction at that point i.e. it turns one of the corners of the polygon.
As z moves along the real axis from the left toward x1, let us assume that w moves along a side of the polygon toward w1. When z crosses from the left side of x1 to the right side, θ1 = arg(z - z1) changes from π to 0 while all other terms in 3] stay constant. Fig. 2a shows the vector z - a1 before z reaches point x1. Fig. 2b shows the
vector z - a1 after z has passed x1. We see that vector z - a1 changes from a left pointing vector to a right pointing vector i.e. changes in direction from π to 0. Consequently arg dw decreases by
(α1/π -1) arg(z - x1) = (α1/π -1)π = α1 - π
or, what is the same thing, increases by π - α1 (an increase being in the counterclockwise direction). As a consequence of this when w reaches point w1, it turns through the angle π - α1 and then moves along the side w1w2 of the polygon. See Fig. 3.
When z moves through point x2 the same thing happens. Arg (z - x2) changes from π to 0 while all other terms in 3] stay constant. Thus another turn through an angle of π - α2 at point w2 is made. We thus see that as z moves along the x axis w traces out the polygon.