1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

490 CHAPTER. 11 • APPLICATIONS OF HARMONIC FUNCTIONS

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Figure 11. 72 The region of interest.

• 'Tr 11
  Using the image values f ( -1) =
  2

and f ( 1) = '2, we obtain the system

• 11 - i11 n in
  2 = A -
  2
  -+B and '2=A-z+B,

which we can solve to obtain B = 0 and A = -i. Hence the required function is

f (z) = Arcsinz.

•EXAMPLE 11. 27 Verify that w = f (z) = (z^2 - l) t maps the upper half-
plane Im (z) > 0 onto the upper half-plane Im (w) > 0 slit along the segment
from 0 to i. (Use the principal square root t hroughout.)

Solution If we choose x 1 = -l,x2 = O,x3 = 1, w 1 = -d,w2 = i , and w3 = d,
then the formula
~ ~ ~

g' (z) = A (z + 1) • (z) • (z - 1) •

will determine a mapping w = g (z) from the upper half-plane Im {z) > 0 onto

the portion of the upper half-plane Im (w) > 0 that lies outside the triangle with

vertices ± d, i as indicated in Figure 11.73(a). If d--+ 0, then w 1 --+ 0, W3 -+ 0,

v v

u

Figure 11.73 The regions of interest.