Review of Complex Variables

The set of complex numbers are numbers that are expressed in the form

where x and y are real numbers, x is called the real  and y is the imaginary part (x=Re{z},y=Im{z}). If y=0, z is a real number and if x=0 then z is a pure imaginary and j is the square root of (-1). Using complex numbers allows us to solve the following equation

x^{2}+1=0

which does not have a solution if we restrict $x$ to be a real number.

Complex Arithmetic:

If a is a positive real number then

Conjugate and Modulus

Definition: The conjugate of a complex number denoted by ( some times denoted by ) is defined as . Note that

.

The modulus (also called absolute value or length) of a complex number is defined as .

Properties

Polar Form

A complex number can be expressed in different forms: (standard, polar and trigonometric)

is the modulus of z and \theta is called the argument of z

Operations in Polar Form:

Some useful relationships:

Euler Formula

DeMoivre Formula

The Poles and Zeros

A complex function F(s) is analytic} in a region if the function and all its derivatives exists in that region. The points at which the function is analytic are called ordinary while points in the s-plane at which the function F(s) are not analytic are called the singular points of F(s). Singular points at which the function F(s) approaches infinity are called the poles of F(s). If F(s) is a rational function then the poles} are the zeros of the denominator polynomial. Points at which F(s) equals zero are called the zeros of F(s). Finite zeros of a rational function F(s) are the zeros of the numerator polynomial.

What are the poles and zeros of the following function

The functions F(s) has two simple poles (at s=-3 and s=-4) and one finite zero (s=-2). It has another zero at infinity. G(s) has double poles (repeated poles) at s=-2 and a simple pole at s=-5 and has no finite zero.

A rational complex function is called proper if . It is strictly proper if