1 Complex Functions
In this section we will define what we mean by a complex func-
tion. We will then generalise the definitions of the exponential,
sine and cosine functions using complex power series. To deal
with complex power series we define the notions of conver-
gent and absolutely convergent, and see how to use the ratio
test from real analysis to determine convergence and radius
of convergence for these complex series.
We start by defining domains in the complex plane. This
requires the prelimary definition.
Definition 1.1
The ε-neighbourhood of a complex number z is the set of com-
plex numbers {w ∈ C : |z−w| < ε} where ε is positive number.
Thus the ε-neigbourhood of a point z is just the set of points
lying within the circle of radius ε centred at z. Note that it
doesn’t contain the circle.
Definition 1.2
A domain is a non-empty subset D of C such that for every
point in D there exists a ε-neighbourhood contained in D.
Examples 1.3
The following are domains.
(i) D = C.
(Take c ∈ C. Then, any ε > 0 will do for an ε-
neighbourhood of c.)
(ii) D = C\{0}.
(Take c ∈ D and let ε = 1
2
(|c|). This gives a
ε-neighbourhood of c in D.)
(iii) D = {z : |z − a| < R} for some R > 0. (Take c ∈ C and let
ε = 1
2
(R− |c− a|). This gives a ε-neighbourhood of c in D.)
Example 1.4
The set of real numbers R is not a domain. Consider any
real number, then any ε-neighbourhood must contain some
complex numbers, i.e. the ε-neighbourhood does not lie in the
real numbers.
We can now define the basic object of study.
1
mohansahutgv@gmail.com
Mohan Sahu
Definition 1.5
Let D be a domain in C. A complex function, denoted f : D →
C, is a map which assigns to each z in D an element of C, this
value is denoted f(z).
Common Error 1.6
Note that f is the function and f(z) is the value of the function
at z. It is wrong to say f(z) is a function, but sometimes people
do.
Examples 1.7
(i) Let f(z) = z2 for all z ∈ C.
(ii) Let f(z) = |z| for all z ∈ C. Note that here we have a
complex function for which every value is real.
(iii) Let f(z) =