11.9 • THE SCHWARZ- CHRJSTOFFEL TRANSFORMATION 491v v(b)
Figure 11. 7 4 The regions of interest.7r 7r
0:1--+ 2,a2--+ -7r, and a3-+ 2 ' The limiting formula for the derivative g'(z)
becomes
- 1 - 1
f' (z) =A (z + 1)^2 (z) (z - 1)^2 ,
which will determine a mapping w = f (z) from the upper half-plane Im (z) > 0
onto the upper half-plane Im (w) > 0 slit from 0 to i as indicated in Figure
11.73(b). An easy computation reveals that f (z) is given by
f(z)=Af zdz =A(z^2 -1)~ +B,
(z2 - l)t
and the boundary values f (±1) = 0 and f (0) = i lead to the solution
f(z)= (z^2 -l)t.
- EXAMPLE 11 .28 Show that the function
!()
w = z = -lA. rcsmz + -iA rcsm-.1 + --l+i
7r 7r z 2
maps the upper half-plane Im (z) > 0 onto the right-angle channel in t he first
quadrant, which is bounded by the coordinate axes and the rays x ~ 1, y = l
and y ~ 1, x = 1, as depicted in Figure l l.74(b).
Solution If we choose x 1 = -1, x 2 = O,x3 = l,w1 = 0, w2 = d, and W3 = l +i,
then the formula
~ ~ ~
g^1 (z) = Ai(z + 1) • (z) • (z -1) •