19MAT214 – Fourier Transform and
Complex Analysis
Complex Numbers
S. Santhakumar
Complex number :
A complex number is a number that can be expressed in the
form z = a + ib, where a and b are real numbers, and i is a solution of
the equation x2 = −1. Because no real number satisfies this
equation, i is called an imaginary number.
For the complex number a + ib, a is called the real part,
and b is called the imaginary part. We can denote it as
Re z = a and Im z = b.
A complex number z, also can be written as an ordered pair
(a, b) of real numbers a and b, written
z = (a, b).
Operations on complex numbers:
The Addition of two complex numbers z1  ( x1 , y1 )
and
z 2  ( x2 , y2 ) is defined by
z1  z 2  ( x1  x2 , y1  y2 )  ( x1  x2 )  i ( y1  y2 )
The multiplication is defined by
z1. z 2  ( x1 x2  y1 y2 , x1 y2  y1 x2 )  ( x1 x2  y1 y2 )  i ( x1 y2  y1 x2 )
The subtraction and division are defined as the inverse
operations of addition and multiplication, respectively.
The difference is the complex number
z1  z 2  ( x1  x2 , y1  y2 )  ( x1  x2 )  i ( y1  y2 )
The quotient is the complex number z
x x  y y   ix2 y1  x1 y2 
z  z1 / z 2  1 2 1 2 2
x1  x22
Complex Plane:
In the XY-plane the horizontal x-axis, will be denote the real
axis, and the vertical y-axis, will be denote the imaginary axis. In this
way every complex numbers can be plotted in XY-plane.
7+i8
6+i6
7+i8
6+i6
Addition:
9+i10
3+i7
6+i3
Subtraction:
6+i5
3+i4
3+i
-3 –i
Laws of addition and multiplication:
Commutative laws:
Commutative
Commutative
law law
for multiplication.
for addition.
Associative laws:
Associative
Associative
lawlaw
forfor
multiplication.
addition.
Distributive law:
Distributive law for addition and
multiplication.
Additive Identity
inverse.
Complex Conjugate:
The complex conjugate z
defined by
of a complex number z = x + iy is
z  x  iy
It is obtained geometrically by reflecting the point z in the real
axis.
5+4i
5-4i
Some Properties of Complex Conjugate :
1.
2.
3.
4.
5.
6.
Polar Form of Complex Numbers:
AbsoluteArgument
value or modulus
(4, 7) = 4+i7
= 8.06 (cos 60.26 + i sin 60.26)
r = 8.06
y = r sin 𝞡
𝞡 = 60.26
x = r cos 𝞡
Triangle Inequality:
Generalized triangle inequality:
For example,
Multiplication and Division in Polar Form :
1. The absolute value of a product equals the product of the
absolute values of the factors
2. The argument of a product equals the sum of the arguments of
the factors
1. Find nth root of the number -1 by using DeMoivre’s Formula.
Soln: The Polar Form of -1 is −1 = cos 𝜋 + 𝑖 sin 𝜋
1
𝑛
−1 = cos
(2𝑚+1)𝜋
𝑛
are the nth root of (-1).
+ 𝑖 sin
(2𝑚+1)𝜋
,
𝑛
𝑚 = 0,1,2, … 𝑛 − 1.
2.
Find 5th root of the complex number 2 − 𝑖2 by using De
Moivre’s Formula.
Soln: The Polar Form of 2 − 𝑖2 is 𝑟, 𝜃 = 2 2, 2𝑛
2 − 𝑖2
1
5
= 2 2
1
5
7
cos
2𝑛+4 𝜋
5
2 2
2 2
1
5
cos
15𝜋
20
cos
11𝜋
20
+ sin
15𝜋
20
+ sin
11𝜋
20
,
𝜋
7
2𝑛+4 𝜋
+ sin
2 2
, 2 2
1
5
n = 0,1, 2,3,4
5
Hence the 5th root of 2 − 𝑖2 are 2 2
1
5
7
+
4
1
5
1
5
7𝜋
20
cos
cos
cos
3𝜋
20
19𝜋
20
+ sin
+ sin
+ sin
7𝜋
20
,
3𝜋
20
19𝜋
20
,
.