COMPLEX ANALYSIS

Complex analysis importance

• Complex analysis has not omly transformed the world of mathematics, but surprisingly, we find its application in many areas of physics and engineering.
• For example, we can use complex numbers to describe the behavior of the electromagnetic field.
• In atomic system, which are described by quantum mechanics, complex numbers and complex function play a central role.

What is a complex number

The Algebra of Complex Numbers

• More general complex numbers can be written down. In fact numbers a and b we can form a complex number: c=a+ib
• We call a the real part of the complex number c and refer to b as the imaginary part of c.

Addition, subtraction, multiplication

Complex conjugate

Complex conjugate

Division is defiened in terms conjugate of the denominator

Graphical representation of complex number

Complex variables

Very Important complex transformations

The Polar Representation

The Polar Representation

The Polar Representation

THE ARGUMENT OF Z

EULER’S FORMULA

Note the similarity

EULER’S FORMULA

EULER’S FORM

EULER’S FORM operations

EULER’S FORM operations

EULER’S FORM operations