1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

64 CHAPTER 2 • COMPLEX F UNCTIONS

are mapped onto points that lie on the ray p > 0, </> = 2a. If we now restrict the

domain of w = f (z) = z^2 to the region

{

A = re' ·o : r > 0 and -'Ir 7r}

2

< (J :S

2

, (2-6)

then the image of A under the mapping w = z^2 can be described by the set

B = {pe•

0 and - 7r < </> :S 7r} , (2-7)

which consists of all points in t he w plane except the point w = 0.

The inverse mapping off, which we denote g , is then

where w E B. That is,

.l 1. Arg(w)

z = g(w) = w• = lwl' e'-,-,

where w f 0. The function g is so important that we call special attention to it

with a formal definition.

I Definition 2.1: Principal sq uare root

The function

I i • A<•(w)

g(w) = w'l = lwl~ e•-,-, for w f 0,

is called the principal square r oot functio n.

(2-8)

We leave as an exercise to show that f and g satisfy Equations (2-3) and

thus are inverses of each other that map the set A one-to-one and onto the set B

and the set B one-to-one and onto the set A, respectively. Figure 2.12 illustrates

this relationship.

What a.re the images of rectangles under t he mapping w = z^2? To find out,

we use the Cartesian form

w = u+iv = f (z) = z^2 = x^2 - y^2 +i2xy = (x^2 - y^2 , 2x y) = (u, v)

and the resulting system of equations

u = x^2 -y^2 and v = 2 xy. (2-9)

• EXAMPLE 2.12 Show that the transformation w = f (z) = z^2 , for z f 0,

usually maps vertical and horizontal lines onto parabolas and use this fact to

find the image of the rectangle {(x, y): 0 < x <a, 0 < y < b}.