Mathematics of Physics and Engineering

190 Functions of a Complex Variable

4.2 Functions of a Complex Variable

4.2.1 Continuity and Differentiability

The study of functions of a complex variable is called complex analysis.
As in the usual calculus, we define a complex function, that is, function
/ = f(z) of a complex variable z as a rule that assigns to every complex
number z at most one complex number f(z). This definition is especially
important to keep in mind: rules that assign several values to the same com-
plex number z also appear in complex analysis and are called multi-valued
functions.

A polynomial function p — p(x) = X3fc=o akxk °f a real variable x
easily extends to the complex plane by replacing a; with z and allowing the
coefficients afc to be complex numbers. Similarly, a rational function,
that is, a ratio of two polynomial functions, extends to the complex plane
except at the points where the denominator vanishes. For other functions of
the real variable, the replacement of x with z is not as straightforward, and
a special theory of the functions of a complex variable must be developed.

Definition 4.1 A function / = f(z) is continuous at ZQ if / is defined
in some neighborhood of ZQ and

lim f(z 0 + z) = f(z 0 ).

\z\—>0

A function / = f(z) is differentiable at ZQ if / is defined in some neigh-

borhood of ZQ and there exists a complex number, denoted by f'(zo), so

that

lim /(«> + *)-/(«>) = f{zo).

EXERCISE 4.2.1.C Verify that a function differentiable at ZQ is continuous
at ZQ.
EXERCISE 4.2.2? Which of the following functions are differentiable at zero:
f(z) = z\f{z) = Z(z)J(z) = \z\J(z) = \z\^2?
EXERCISE 4.2.3.c Verify that if f(z) = zk for a positive integer k, then
f'{z) = kzk-^1.
Definition 4.2 A function is called analytic at a point if it is differ-
entiable in some neighborhood of the point. A function is called analytic
in a domain if it is analytic at every point of the domain. In general, we