1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

414 CHAPTER 10 • CONFOR.MAL MAPPING

y

_____!

-2 -I 0

Z=t2-J\

2

"'=f(z)

--

-2

y

• I

~-~

v

0 2

/=zt

x

Figure 10.13 The composite transformation w = f (z) = (z^2 - 1)! and the interme-

diate steps z = z^2 - 1 a.nd w = z~.

1

10.3.1 The Mapping w = (z^2 - 1)2

The double-valued function f (z} = (z^2 - 1) t has a branch that is cont inuous

for values of z dist ant from the origin. This feature is motivated by our desire

1

for the approximation ( z^2 - 1) ~ ~ z to hold for values of z distant from the

t

origin. We begin by expressing (z^2 - 1)2 as

1 1

w = fi(z) = (z -1)2 (z + 1)^2 , ( 10-22)

where the principal branch of the square root function is used in both factors.

We claim that the mapping w =Ji (z) is a one-to-one conformal mapping from

the domain set D 1 , consisting of the z plane slit along the segment - 1 ~ x ~ 1,

y = 0, onto the range set H 1 , consisting of thew plane slit along the segment

u = 0, - 1 ~ v ~ l. To verify this claim, we investigate the two formulas on the

right side of Equation (10-22) and express them in the form