1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

66 CHAPTER 2 • COMPLEX FUNCTIONS

y

v

2 ib a+ib

Figure 2.13 T he transformation w = z^2.

What happens to images of regions under the mapping

1 ·~ l · t ·9

w =f(z)=lzl'e'^2 =r>e'> forz= re' :;{:O,

where -1f < (J :S 7r? If we use polar coordinates for w = pei in the w plane, we

can represent this mapping by the system

p=r~ and ¢=~.
2

(2-12}

Equations (2-12) indicate that the argument off (z) is half the argument of

z and that the modulus off (z) is the square root of the modulus of z. Points

that lie on the ray r > 0, (J = a are mapped onto the ray p > 0, </> = ~. The

image of the z plane (with the point z = 0 deleted) consists of the right half·

plane Re (w) > 0 togethe r with the positive v-axis. The mapping is shown in

Figure 2.14.

v

x u

p:rl

9 =~

-rr< es "

Figure 2.14 T he mapping w = z!.