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Complex Analysis: Balram Dubey

The document provides information about complex numbers including: 1. A complex number z is represented as an ordered pair (x, y) where x is the real part and y is the imaginary part. Complex numbers are often written as z = x + iy. 2. Important operations on complex numbers include addition, multiplication, and division. 3. The complex plane is used to represent complex numbers graphically with the real part on the x-axis and imaginary part on the y-axis. 4. Equality of complex numbers, properties of operations, and conjugate of a complex number are discussed. Polar form, modulus, and argument of a complex number are also introduced.

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Kishan Panpaliya
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184 views57 pages

Complex Analysis: Balram Dubey

The document provides information about complex numbers including: 1. A complex number z is represented as an ordered pair (x, y) where x is the real part and y is the imaginary part. Complex numbers are often written as z = x + iy. 2. Important operations on complex numbers include addition, multiplication, and division. 3. The complex plane is used to represent complex numbers graphically with the real part on the x-axis and imaginary part on the y-axis. 4. Equality of complex numbers, properties of operations, and conjugate of a complex number are discussed. Polar form, modulus, and argument of a complex number are also introduced.

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Kishan Panpaliya
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COMPLEX

ANALYSIS

Balram Dubey
TEXT BOOK:
• Complex Variable & Applications

• 8th Edition

• Authors: James Ward Brown


Ruel V. Churchill
Complex Number: A complex
number z is an ordered pair (x, y),
where x & y are real nos. i.e.
z = (x, y), where
x = real part of z = Re z
y = imaginary part of z = Im z
We usually write

z= (x, y) = x + i y,

where i =
1 = (0, 1)

i2 = i. i = (0, 1) . (0, 1) = ( -1, 0)


• (WAIT !)
Important Operations

1.Addition of complex numbers:

z1 + z2 = (x1+iy1) +(x2+iy2)

=(x1+x2)+i(y1+y2)
2. Multiplication of complex
numbers:
z1 z2 = (x1+iy1) (x2+iy2)

=(x1x2- y1y2)+i(x1y2+x2y1)
3. Division:
If z1  x1  iy1 &
z2  x2  iy2  0  i.0, then
z1 x1  iy1 x1  iy1 x2  iy2
z   
z2 x2  iy2 x2  iy2 x2  iy2
x1 x2  y1 y2 x2 y1  x1 y2
 i 2
x2  y2
2 2
x2  y2 2
Complex Plane :
y

x P z= (x, y)

Imaginary
axis 1 yx

O
1

Real axis
Complex Plane:

• Choose the same unit of length on


both the axes

• Plot z = (x, y) =x +iy as the point P


with coordinates x & y.
• The xy-plane, in which the
complex nos. are represents in
this way, is called complex
plane or Argand diagram .
Equality of two complex nos:
Two complex nos. z1 & z2 are said to
be equal iff
Re (z1) = Re (z2) &
Im(z1) = Im(z2).
Properties of Arithmetic operations:

(1) Commutative Law:

z1+z2 = z2+z1

z 1z 2 = z 2z 1
2. Associative law:

(z1+z2)+z3=z1+(z2+z3)
(z1z2) z3 = z1 (z2z3)

3. Distributive law

z1(z2 + z3)= z1z2 + z1z3


(z1+z2)z3= z1z3 + z2z3
4. z + (-z) = (-z) +z = 0

5. z.1 = z
• Complex conjugate number:

Let z = x+iy be a complex


number.
Then z = x–iy is called
complex conjugate of z
Properties of complex nos.:
1. z  z  2 x
1
 x  Re z  ( z  z )
2
1
2. y  Im z  ( z  z )
2i
3. z1  z 2  z1  z 2

4. z1 z 2  z1 z 2

 z1  z1
5.   
 z2  z2
6. z  z
7. z is real iff z  z.
8. iz  i z   i z
9. Re (iz )   Im ( z ), iz  ix  y
10. Im (iz )  Re( z )
11. z1 z 2  0  z1  0 or z 2  0
Polar Form of complex Numbers:
Let z = x+iy
Put x = r cos, y = r sin
z = r (cos + i sin ) = r ei
which is called polar form of
complex number.
MODULUS OF COMPLEX NUMBER

z r  x y 0
2 2

Geometrically, z is the distance


of the point z from the origin.
Y

y
P z=(x+ iy)


x X
O
 z1  >  z2 means that the point z1 is
farther from the origin than the point z2.
  z1-z2  = distance between z1& z2

z2
ARGUMENT OF COMPLEX NUMBER

The directed angle  measured from


the positive x-axis is called the
argument of z, and we write  = arg z.

z = x+iy


• Remarks :

1. For z = 0,  is undefined.
2.  is measured in radians, and is
positive in the counterclockwise sense.
3.  has an infinite number of possible
values, that differ by integer multiples of
2. Each value of  is called argument
of z, and is denoted by  = arg z
4. When  is such that - <   , then

such value of  is called principal value of

arg z, and is denoted by

Θ = Arg z, if -  < Θ  

5. arg z= Arg z + 2n, n = 0,  1, 2,……..


i1 i 2
6. Let z1  r1e , z 2  r2 e .
Then z1  z 2  (i ) r1  r2 &
(ii ) 1   2  2n
n  0,  1,  2,.....

7. arg( z1 z 2 )  arg( z1 )  arg( z 2 )


How to find argz / Argz ?
Ex1. Let z  1  i, Argz  ?
Sol :
We have
z  1  i  r (cos  i sin  )
 z r 2
 1  i  2 (cos  i sin  )
 2 cos  1, 2 sin   1
 tan   1
     Argz  3 / 4
Hence
arg z  Argz  2n , n  0,1,2,..
 (3 / 4)  2n , n  0,1,2,..
Ex2. Let z  2i, Argz  ?
Sol :
We have
z  2i  r (cos  i sin  )
 z r2
 2i  2(cos  i sin  )
 2 cos  0, 2 sin   2
     Argz   / 2
Hence
arg z  ( / 2)  2n , n  0,1,2,..
Roots of Complex Numbers:
For z0  0, there exists n values of
z which satisfy z  z0 n

i n in 
Let z  re  z  r e n

i 0
Let z  z0  r0 e
n
, n  2, 3,.....
n in  i 0
Then r e  r0 e
 r  r0 , n

n   0  2k ,
 0  2k
 r  (r0 ) 1/ n
, 
n
i
z  r e
1  0  2 k
i( )
 z  zk  (r0 ) n e n

is called nth roots of z0 , k  0,1,.., n  1.


Principal Root.

For k  0,
i 0 / n
z0  (r0 )
1/ n
e

is called the PRINCIPAL ROOT.


Triangular inequality:
1. z1  z 2  z1  z 2

2. z1  z2  z1  z2

3. z1  z2  z1  z2

4. z1  z2  z1  z2
Let z = x+iy, Then z is the

distance of the point P (x,y)

from the origin


Y
OP=z

O x
If z1  x1  iy1 and z2  x2  iy 2 ,
then z1  z2  distance between z1 & z2 .

z2
z1  z2
Let C be a circle with centre z0 and

radius . Then such a circle C can be

represented by C:z-z0=  .

c
z0 z-z0= 
Consequently, the inequality

z-z0 <  ----------(1)

holds for every z inside C.

i.e. (1) represents the interior of C.


Such a region, given by (1), is
called a neighbourhood (nbd) of
z0, i.e. the set
N(z0) ={z: z-z0< }
is called a nbd. of z0
Deleted neighborhood:

N0 = {z: 0 < z-z0<  } is called


deleted nbd.
It consists of all points z in an
 -nbd of z0, except for the point
z0 itself.
• The inequality z-z0>

represents the exterior of the

circle C.
Interior Point:

Let S be any set. Then a point z0S is

called an interior point of S if  a nbd

N(z0) that contain only points of S, i.e.

z0 N(z0)  S
Exterior Point: A point z0 is called an

exterior point of the set S if  a nbd N

of z0 that contains no points of S.

z0 is an ext. pt. of S  z0 is an int. pt

of Sc.
Boundary point:

A point z0 is called boundary point

for the set S if it is neither interior

point nor exterior point of S.


Open Set:
A set S is said to be open if every

point of S is an interior point of S, i.e.

S is open iff it contains none of its

boundary points.
Closed set:
A set S is said to be closed if its

complement Sc is open, i.e. S is

closed iff it contains all of its

boundary points.
Closure of a set:
• Closure of a set S is the closed set
consisting of all points in S together with
the boundary of S.

Ex1. Let S  {z : z  1}.


Then Cl ( S )  {z : z  1}.
Ex2. Let S  {z : z  1}.
Then Cl ( S )  {z : z  1}.
Bounded set:
A set S is called bounded if all of

its points lie within a circle of

sufficiently large radius, otherwise

it is unbounded.
Connected Set:
An open set S is said to be
connected if any of its two points
can be joined by a broken line of
finitely many line segments, all of
whose points belong to S.
• Q. Is the set

S  {z : z  1}  {z : z  2  1}
connected ?
Domain:

An open connected set is called a

domain.
Ex1: Sketch & determine which are domains

(a)S = {z:  z-2+i 1}

We have z-2+i 1

 x+iy -2+i 1

(x-2)+i (y+1) 1

(x-2)2 + (y+1)21

(2,-1)
 S contains the interior &

boundary pts. of a circle with centre

(2, -1) & radius 1.

 (i) S is not a domain

(ii) S is bounded.
Ex2. S = { z:2z+3>4}

We have 2z+3>4

2x+3+ i 2y >4

 (2x+3)2 +4y2 >16

 (x+3/2)2 +y2 >4


• Clearly S contents the exterior
3
pts of a circle with centre ( ,0) &
2
radius 2.

•S is a domain and it is

unbounded
 z 1 
Ex. 3 S  z :  1
 z 1 
Sol. Note that : z  1  z - 1

 z  1  z -1
2 2

 (z  1)( z  1)  (z - 1)( z - 1)
 x  0.
S is a domain and it is unbounded.
END

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