Complex Analysis

Prepared by

Dr. Taha MAhdy

Complex analysis importance

• Complex analysis has not only 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 systems, which are described by quantum mechanics, complex numbers and complex functions play a central role,

What is a complex number

• It is a solution for the equation

The Algebra of Complex Numbers

• More general complex numbers can be written down. In fact, using real 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

• The complex conjugate is:

• Note that

Complex conjugate

Division is defiened in terms of conjugate of the denominator

Graphical representation of complex number

Complex Variables

• A Complex Variable can assume any complex value

• We use z to represent a complex variable.

z = x + jy

• We can graph complex numbers in the x-y plane, which we sometimes call the complex plane or the z plane.

• We also keep track of the angle θ that this vector makes with the real axis.

Very Important complex transformations

It appears that complex numbers are not so “imaginary” after all;

The Polar Representation

• Let z = x + iy is the Cartesian representation of a complex number.

• To write down the polar representation, we begin with the definition of the polar coordinates (r,θ ):

x = r cosθ ; y = r sinθ

The Polar Representation

The Polar Representation

• Note that r > 0 and that we have

• tanθ = y / x as a means to convert between polar and Cartesian representations.

• The value of θ for a given complex number is called the argument of z or arg z.

THE ARGUMENT OF Z

EULER’S FORMULA

• Euler’s formula allows us to write the expression cosθ + i sinθ in terms of a complex exponential.

• This is easy to see using a Taylor series expansion.

• First let’s write out a few terms in the well-known Taylor expansions of the trigonometric functions cos and sin:

Note the similarity

EULER’S FORMULA

EULER’S FORM

• These relationships allow us to write a complex number in complex exponential form or more commonly polar form. This is given by

EULER’S FORM operations

EULER’S FORM operations

EULER’S FORM operations

DE MOIVRE’S THEOREM

Assignment

• Solve the problems of the chapter