Complex Numbers, Complex Functions and Contour Integrals

2. Complex Functions

Now that we’ve extended our collection of numbers to the field of complex numbers, we can begin studying
complex-valued functions. The attempt to extend the theory of calculus to such functions is the branch of
mathematics known as complex analysis. In the complex world, both differentiation and integration gain new
depth and significance. As a sacrifice, however, we must severely limit the class of functions we can analyze,
restricting to so-called analytic (or holomorphic) functions. For a rough definition, suppose f(x, y) is any
complex-valued function of two real variables. Let z = x + iy. We call z the complex variable. By Exercise 3,
we can use the identities and to express f(x, y) as a function of z and Morally,
the function f is analytic if it does not depend on The analytic functions are exactly the complex functions
for which a derivative exists.

Exercise 4. Find the roots of f(z) = z^4 − iz^3 + 2i.

It is a fact that all complex polynomials are analytic. We can use polynomials to construct more general
complex functions. For instance, consider a quotient of two complex polynomials, say f(z) = p(z)/q(z) . For simplicity,
assume p and q have distinct roots. It is then a fact that f is analytic away from the roots of q, which we call
the poles of f. More generally, we have the following definition:

Definition. Suppose f is a function which is analytic in a neighborhood of a point a, except perhaps at a itself.
If then the point a is said to be a pole of f.

By the following fact, the nature of a general pole is exactly the same as in the case of a rational function:

Fact. If f has a pole at a, then there exists a positive integer n and analytic function g not vanishing at a such
that

The integer n is called the order of the pole. Poles of order one are called simple poles. A function which is
analytic away from a discrete set of poles is a called meromorphic. A function which is analytic everywhere is
sometimes called entire.

Exercise 5. What are the poles of

Associated to every pole of a meromorphic function f is a special number called the residue of f at that pole.

Definition. Suppose f is a meromorphic function with a pole at a. The residue of f at a is the unique complex
number R such that

is the derivative of an analytic function in a neighborhood of a, excluding a itself. We’ll usually denote this
number by .

This definition may seem rather arbitrary, but in Section 3 we’ll see that residues are intimately linked with
integration.

Exercise 6. Find the poles and corresponding residues of the function