1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

72 CHAPTER 2 • COMPLEX F UNCTIONS

y v .,,..,,..---- -

// .... ' .....

w=f(z)

Figure 2. 17 The limit f (z)-+ Wo as z-+ zo.

J Definition 2.4: Limit of f (z)

' \ \

\ I

I

I

The expression Jim f (z) = wo means that for each real number e > 0, there

z - zo

exists a real number 8 > 0 such that

If (z) - wol < c whenever 0 < lz - zol < 8.

Using Equations (1-49) and (1-51), we can also express the last relationship as

f (z) E De (wo) whenever z E D6 (zo).

The formulation of Limits in terms of open disks provides a good context

for looking at this definition. It says that for each disk of radius c about the

point wo (represented by D, (wo)) there is a punctured disk of radius 8 about

the point zo (represented by D6 (zo)) such that the image of each point in the

punctured 8 disk lies in the c disk. The image of the 8 disk does not have to fill

up the entire e disk; but if z approaches z 0 along a curve that ends at z 0 , then

w = f (z) approaches wo. The situation is illustrated in Figure 2.17.

• EXAMPLE 2.16 Show that if f (z) = z , then lim f (z) = zo, where zo is

z- zo

any complex number.

Solution As f merely reflects points about the y-axis, we suspect that any c

disk about the point zo would contain the image of the punctured 8 disk about

zo if 8 = c. To confirm this conjecture, we let c be any positive number and

set 8 = e:. Then we suppose that z E D6 (zo) = D; (zo), which means that

0 < iz -zol < e. The modulus of a conjugate is the same as the modulus of

the number itself, so the last inequality implies that 0 < lz -zol < e:. This is

the same as 0 < lz -zol < e. Since f (z) = z and wo = zo, this is the same as

0 < !f(z) -wol < c, or f(z) E De (wo), which is what we needed to show.