1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

494 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

as shown in Figure 1 1.79. Hint: Set xi= - l,x2 = O,xs = l,w 1 = i1r-d,W2 = i;,

and ws = -d and let d _, oo.

u

Figure 11. 79

6. Show that w = f (z) = --2 z (1 - z^2 ) -! - Ar^2 csin z maps t he upper half-plane onto

1r 1r

t he domain indicated in Figure 11.80. Hint: Set x1 = -l,x2 = l,w 1 = 1, and

w2 = -1.

v

Figure 11. 80

7. Show that w = f (z) = z+ Logz maps the upper hal f-plane Im (z) > 0 onto the
   upper half-plane Im(w) > 0 slit along the ray u:::; -1,v ='Ir, as shown in Figure
   11.81. Hint: Set x 1 = -l,x2 = O,w 1 = - 1 + i'lr, and w2 = - d and let d _, oo.
   v

u

Figure 11. 81

.1. 1 -(z+l)!

8. Show that w = f (z) = i1r+2 (z + 1) ~+Log! maps the upper half-plane
   l+(z+l )
   onto the domain indicated in Figure 11.82. Hint: Set x 1 = -1, x2 = 0, w 1 = i'lr,
   and W2 = -d and let d _, oo.
   v

u

Figure 11.82