TABLE OF CONTENTS In this chapter we lay the proper foundations for the exponential and logarithmic functions, for trigonometric functions and their inverses. We calculate derivatives of these functions and use this for establishing important properties of these functions.
We have already mentioned that some theorems from the previous chapter are valid for differentiation in the complex domain. Specifically, this is so for Theorem 13.1 and 13.2, for basic rules of differentiation in Theorems 13.3, 13.5 and for the chain rule, Theorem 13.4. In contrast, Theorem 13.7 makes no sense in the complex domain since the concept of an increasing function applies only to functions which have real values. However, part (iii) of Theorem 13.7 can be extended as follows.
For a ? 
and R >0 denote by S the disc { z; z ?a . If f'(z)=0 for all z ? S then f is constant in S.
For t ?[0, 1] let Z=tz 1+(1 ? t) z 2. If z 1 and z 2 are in S so is Z. Let F: t 
f(Z), F 1 (t)= 
F(t) and f 2 (t)= 
F(t). Then
Consequently 
. By (iii) of Theorem 13.7 both F 1 and F 2 are constant in [0, 1], hence f( z 1)= f( z 2) and f is constant in
