Page 23 - Calculus with Complex Numbers

C o m p I ex f u n c t i o n s

2. I Polynom ials
Having constnlcted the complex number system the next task is to corlsider how
the standard functions we do real calculus with extend to complex variables. Poly-
nomials cause no problems since they only require addition multiplication and
z
subtraction for their delinition. For example, p (z) = 3.: + 4, q (z) = 4z2 - 5. + 6,
etc. The numbers occurring are called coeficients. The degree of the polynomial
is the highest power of z occurring with a non-zero coeflicient.

2.2 Rational fu nctions

These are functiorls of the form r (z) = p (z)(q (z) where p (z), q (z) are polyno-
mials. They can be delined for all z except where the denominator vanishes. Such
points are called singularities. Hvel'y rational function has at least one singularity
because of the fundamental theorem of algebra. For example

z + 1
r(z) =
z + 2
has a singularity at z = -2, whilst

z2 + 1
.
, V(z) =
. z2 + 4
has two singularities at z = +2/ .

2 . 3 G rap h s
Hvel'y real function y = f @) of a real variable .x has a graph in two dimensional
space. For example, Figure 2.1 shows the graph of y = .x2 .
For a complex function w = f (z) of a complex variable z this option is not avail-
able because the graph is a two-dimerlsional surface in a four-dimensional space.
W hat we have to do instead is to draw two diagrams which we call a z-plane and