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 wellknown 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