1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.9 • THE SCHWARZ-CHRISTOFFEL TRANSFORMATION 493

2. Let a be a real constant. Use the Schwarz-Chru.'tOffel formula. to show that the
   function w = f (z) = Log (z -a) maps the upper half-plane Im (z) > 0 onto the
   infinite strip 0 < v < ,,. shown in F igure 11. 76. Hint : Set x 1 = a -1, x 2 = a , w 1 =
   i7r, and w2 = -d and let d--> oo.

v

II

Figure 11 .76

In Exercises 3-15, construct the derivative f'(z) a.nd use the Schwarz- Christoffel
formula, Equation (11-40), and techniques of integration to determine the re-
quired conformal mapping w = f (z).

3. Show that w = f(z) =; (z^2 - l) t +;Log [z+ (z^2 - l)t]-i maps t he upper
   half-plane onto the domain indicated in Figure 11.77. Hint: Set x1 = -1, x2 =
   l,w 1 = 0, and w2 = - i.

v

II

Figure 11.7 7

4. Show that w = f (z) = ~ (z^2 - 1) ~ + ~Arcsin! maps the upper half-plane onto

7r 7r z

t he domain indicated in Figure 11.78. Hint: Set x 1 = w 1 = -l,x2 = O, xs = ws =

1, and w2 = -id and let d-> oo.

v

u

Figure 11.78

5. Show that w = f (z) = ~Log (z^2 - 1) = Log [ (z^2 - 1) t] maps the upper half-
   plane Im ( z) > 0 onto the infinite strip 0 < v < 7r slit along the ray u ~ 0, v = ~,