1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

158 CHAPTER 5 • ELEMENTARY FUNCTIONS

y v

41t

3

21t

1t

".

..
.

• 2 - 1 l • 2

x

Figure 5.1 T he points {zn} in the z plane (i.e., the xy plane) and their image wo =
exp (Zn) in thew plane (i.e., the uv plane).

same point in the w plane. Thus, the complex exponential function is periodic
with p eriod 27ri, which establishes Equation (5-2). We leave the verification of
Equations (5-3) and (5-4) as exercises.

• EXAMPLE 5 .1 For any integer n, the points

Zn=4^5 +i. (ll?r 6+2n7r )

in the z plane are mapped onto the single point

w 0 = exp(Zn)=e• • ( cosll?r.. ll?r)
6

+ism6

.J3 ~ , 1 ~

= - e• - i - e•

2 2

~ 3.02 - 1.75i

in thew plane, as indicated in Figure 5 .L

Let's look at t he range of the exponential function. If z = x + iy, we see
from Identity (5-1)-e• = e"'eiv = e" (cosy+ isiny)-that e• can never equal
zero, as ex is never zero, and the cosine and sine functions are never zero at the
same point. Suppose, then, that w = e • =/; 0. If we write w in its exponential
form as w = pei, Identity (5-1) gives

Using Identity (5-1), and Property (1-41) of Section 1.5, we get

p = e"' and if!= y + 2n?r, where n is an integer. Therefore,

P = le'I = e"', and

if! E arg (e') = {Arg (e') + 2n7r : n is an integer}.

(5-5)

(5-6)

(5-7)