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19.2 Complex Functions 677

In this section, we will extend the calculus concepts of continuity and differentiability to com-
plex functions and also develop complex versions of powers, exponentials, logarithms, and
trigonometric functions.

19.2.1 Limits, Continuity, and Differentiability

If f is a complex function, f (z) has limit L as z approaches z 0 if, given any positive
number , there is a positive number δ such that
| f (z) − L| <
for all z in S such that 0 < |z − z 0 | <δ.

This means that we must be able to make the values f (z) as close as we like to L by
confining z to a small enough disk about z 0 , excluding the center z 0 . The actual value of f (z 0 ),
if this is defined, is not relevant for the limit, which has to do only with the behavior of f (z) as
z is taken close to z 0 .

EXAMPLE 19.7
Let f (z) = z for z = i. Even though f (i) is not defined,
2
lim f (z) = i =−1.
2
z→i
A significant difference between limits of complex and real functions is that (on the real line)
x can approach x 0 only from the left or right, while in the complex plane, z can approach z 0 along
infinitely many different paths (Figure 19.6). Requiring that f (z) approach the same number L
along all such paths is a much stronger condition than requiring that the function approach the
same value only from the left or right.
As in calculus, we rarely invoke the − δ definition of limit to evaluate a limit. The limit
of a finite sum (product, quotient) is the sum (product, quotient) of the limits whenever all
the limits are defined, and in the case of a quotient, the denominator is nonzero. Furthermore,
cf (z) = cL for any number c.
lim z→z 0 f (z) = L implies that lim z→z 0
f (z) = f (z 0 ), we say that f is continuous at z 0 . This requires
In the special case that lim z→z 0
that f (z 0 ) be defined and that f (z) approach f (z 0 ) as z approaches z 0 along any path. The
function of Example 19.7 is not continuous at i because f (i) is not defined.

y

x

z 0 z

FIGURE 19.6 z → z 0 along arbitrary
paths.

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October 15, 2010 18:5 THM/NEIL Page-677 27410_19_ch19_p667-694