1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

408 CHAPTER 10 • CONFORMAL MAPPING

y v

w = S(z)

-

x

Figure 10.7

. -iz + 4i

The mappmg w = S (z) =.

z

which determines a mapping of the disk lz - 21 < 2 onto the upper half-plane

Im ( w) > 0. We simplify the preceding equation to obtain the desired solution:

S()

• iz+4i
  W= Z =.
  z

A straightforward calculation shows that the points z 4 = 1 - i, z 5 = 2, and

Z6 = 1 + i are mapped onto t he points

W4 = 8 (1 -i) = -2 + i, ws = S (2) = i, and w6 = S (1 + i) = 2 + i,

respectively. The points W4, w5, and w6 lie on the horizontal line Im (w) = 1 in

the upper half-plane. Therefore, the crescent-shaped region is mapped onto the

horiwntal strip 0 <Im (w) < 1, as shown in Figtire 10.7.

10.2.1 Lines of Flux

In electronics, images of certain lines represent lines of electric flux, which com-

prise the trajectory of an electron placed in an electrical field. Consider the

bilinear t ransformation

z aw

w=S(z)=-- and z=S-^1 (w)=- -.

z - a w - 1

The half-rays { Arg ( w) = c}, where c is a constant, that meet at the origin

w = 0 represent the lines of electric flux produced by a source located at w = 0

(and a sink at w = oo). The preimage of this family of lines is a family of circles

that pass t hrough the points z = 0 and z = a. We visualize these circles as

the lines of electric flux from one p oint charge to another. The limit ing case as

a--+ 0 is called a dipole and is discussed in Exercise 6, Section 11 .11. The graphs

for a = 1, a= 0.5, and a= 0.1 are shown in Figure 10.8.