I. Some preliminaries on complex numbers.

1. Polar forms of complex numbers:
 

Let 

then

where r is known as the magnitude of z, denoted |z|, and q is known as the argument of z, denoted arg(z)
Thus the polar form for z is given by where,

2. Some properties of complex numbers:

Let conj(z) be the conjugate of z, e.g. conj(a+ i b)=a - i b
a)

b)

3. Functions of complex numbers yield complex numbers

4. A function of the conjugate is the conjugate of the function

Example: Let

then

Example: Let

then

II. Frequency Response of a stable Transfer Function
 

Suppose we are given a transfer function  G(s)  from u(s) to y(s), whose poles all have negative real parts. Using a sinusoidal input, u(t)=A sin(wt), the Laplace transform is given by

The output, y(s) in the Laplace domain is given by

For discussion purposes, we will use a second order process (with b,c>0)

then

After separation into partial fractions,

Using the short cut method to determine a and b,

which yields

Thus,

As t goes large, the transients die out - leaving only the last two terms,
[Equation 1]:

Based on earlier discussions, this was put in the form,
[Equation 2]:

Comparing equation 1 with equation 2,

After some more rearrangements,

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