Chap 1: Complex Numbers & Complex Functions
I. Complex Numbers

A complex number is an ordered pair of real number :
Definition
The real part:
The imaginary part:
The imaginary unit: , where
If is said to be purely imaginary
If is said to be purely real
VII. Complex Sequences & Series:
Limit Laws (because series so we only have )
Basic Rule
If and are continuous functions and is a constant:
Squeeze
Theorem

L Hospital
Rule

The radius of convergence:
Given a complex sequence , the complex series or is:
Chap 1: Complex Numbers & Complex Functions
1. Complex Numbers
 A complex number is an ordered pair of real number :
Definition
The real part:
The imaginary part:
The imaginary unit: , where
If is said to be purely imaginary
If is said to be purely real
Chap 1: Complex Series
1. Complex Sequences & Series
Limit Laws (because series so we only have )
Basic Rule
If and are continuous functions and is a constant:
Squeeze
Theorem

L Hospital
Rule

The radius of convergence:
Given a complex sequence , the c
Chap 2: Laplace Transform
1. The Laplace Transform
a. Laplace Transform
 The Laplace transform of a function is:
Definition &
Notation
Existence of
Laplace
Transform
Example

is the complex variable. The domain of is called frequency domain
is the real
Chap 3: Z Transform
1. The Z Transform
a. Z Transform
 Infinite sequence:
 Causal sequence:
 The z Transform of a causal sequence is defined whenever the sum exists and where is a complex variable:
Definition

The Z Transform of an infinite sequence:
Chap 4: Fourier Series
1. Fourier Series Expansion
a. Fourier Series Expansion of functions of period T (general case)
 A periodic function has its image values repeated at regular intervals in its domain:

The period of the function is the interval bet
Chap 4.2: Fourier Transform
1. The Fourier Transform
a. The Fourier Integral

If is an odd function then we have the Fourier sine
integral:
Fourier integral may be written in the form of the pair of
equations

The Fourier transform of the function :

T