1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11 .9 • THE SCHWARZ-CHRISTOFF.EL TRANSFORMATION 495

9. Show that w = f (z) = (z - l)" [l + az/ (1 - a))^1 - " maps the upper half-plane
   Im (z) > 0 onto the upper half-plane Im (w) > 0 slit along the segment from 0 to
   e'" ~, as shown in Figure 11.83.
   Hint: Show that/' (z) =A [z + (1-a) far"' (z) (z -1)"-^1.

Figure 11.83

' (z+1)i- 1 i - (z+l)t
10. Show that w = f (z) = 4 (z + l)" +log 1 + ilog l maps

(z+l)i +l i+(z+l)

the upper half-plane onto the domain indicated in Figure 11.84. Hint: Set z1 =

• 1, z2 = 0, w 1 = i1T, and W2 = - d and let d-+ co. Use the change of variable
  z + 1 = s^4 in the resulting integral.
  v

u

Figure u.84

• i i

11. Show that w = f (z) = -z• (z - 3) maps the upper half-plane onto the domain
    z
    indicated in Figure 11.85. Hint: Set x 1 = 0, x2 = 1, w 1 = -d, and w2 = i and let
    d-> o.
    v

"

Figure 11.85

J

dz

12. Show that w = f (z) = 3 maps the upper half-plane Im (z) > 0 onto a
    (1 -z2)•
    rig ' ht tnang. I e wit "h ang I es 1T 2, ... 4 , an d 4 · ...


13. Show that w = f (z) = J dz 1 maps the upper half-plane onto an equilateral
    ( l - z^2 ) •
    triangle.