7 Elementary Props Of Complx Nos.

Complex Number: A complex number z is an ordered pair (x, y), where x & y are real nos. i.e. z = (x, y) x = real part of z = Re z − 1 z = Im z y = imaginary part of

We usually write z= (x, y) = x + i y, where i =

−1

= (0, 1)

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

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 & z 2 = x2 + iy2 ≠ 0 + i.0, then z1 x1 + iy1 x1 + iy1 x2 − iy2 z= = = × z 2 x2 + iy2 x2 + iy2 x2 − iy2 x1 x2 + y1 y2 x2 y1 − x1 y2 = +i 2 2 2 2 x2 + y 2 x2 + y 2

Complex Plane: • Choose the same unit of length on both the axis • 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 .

Complex Plane : y

x

Imaginary axis

P z= (x, y)

yx

1

O

1 Real axis

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 z1z2 = z2z1

1. 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 – i y 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

5.

 z1  z1   = z  2  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 cosθ, y = r sinθ ∴z = r (cosθ + i sin θ) = r eiθ 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)

r =  z 

θ

O

x

X

 z1  >  z2 means that the point z1 is farther from the origin than the point z2.   z1-z2  = distance between z1& z2 z

z2 z

2



 -z 2

1

z1 z 1

ARGUMENT OF COMPLEX NUMBER The directed angle θ measured from the positive x-axis is called the argument of z, and we write θ = arg z.

z = x+iy θ

• Remarks : 1. For z = 0, θ is undefined. 2. θ is measured in radians, and is positive in the counterclockwise sense. 3. θ has an infinite number of possible values, that differ by integer multiples of 2π. Each value of θ is called argument of z, and is denoted by θ = arg z

4. When θ is such that -π < θ ≤ π, then such value of θ is called principal value of arg z, and is denoted by Θ = Arg z, if - π < Θ ≤ π

5. arg z= Arg z + 2nπ, n = 0, ± 1, ±2,……..

i θ1

iθ 2

6. Let z1 = r1e , z2 = r2e . Then z1 = z2 ⇔ (i ) r1 = r2 & (ii ) θ 1 = θ 2 + 2nπ n = 0, ± 1, ± 2,..... 7. arg( z1 z2 ) = arg( z1 ) + arg( z2 )

How to find argz / Argz ?

Ex1. Let z = −1 + i,

Argz = ?

Sol : We have z = −1 +i = r (cos θ +i sin θ) ⇒ z =r = 2 ∴−1 +i = 2 (cos θ +i sin θ)

⇒ 2 cosθ = − 1, ⇒ tan θ = − 1

2 sin θ = 1

⇒ θ = Θ = Argz = 3π / 4 Hence arg z = Argz + 2nπ , n = 0,± 1,± 2,..

Ex2. Let z = −2i, Argz = ? Sol : We have z = − 2i = r (cos θ + i sin θ ) ⇒ z =r=2 ∴ − 2i = 2(cos θ + i sin θ )

⇒ 2 cosθ = 0, 2 sin θ = − 2 ⇒ θ = Θ = Argz = −π / 2 Hence arg z = (−π / 2) + 2nπ , n = 0,± 1,± 2,..

Roots of Complex Numbers:

For z0 ≠ 0, there exists n values of n

z which satisfy z = z0 iθ

n inθ

n

Let z = re ⇒ z = r e n

Let z = z0 = r0e n inθ

Then r e

iθ 0

= r0e

iθ 0

, n = 2, 3,.....

n

⇒ r = r0 , nθ = θ 0 + 2kπ , 1/ n

⇒ r = (r0 ) ∴z = r e

θ 0 + 2kπ ,θ = n

iθ

⇒ z = z k = (r0

1 )n

θ 0 + 2 kπ ) i( n e

is called nth roots of z0 , k = 0,1,.., n − 1.

Principal Root.

For k = 0, 1/ n

z0 = (r0 )

e

iθ 0 / n

is called the PRINCIPAL ROOT.

Triangular inequality: 1. z1 + z 2 ≤ z1 + z 2 2. z1 − z 2 ≤ z1 + z 2 3. z1 + z 2 ≥ z1 − z 2 4. z1 − z 2 ≥ z1 − z 2

Let z = x+iy, Then z is the distance of the point P (x,y) from the origin Y

y

i + x

= z ,

 z 

O

OP=z

P

x

If z1 = x1 + iy1

and z 2 = x2 + iy2 ,

then z1 − z 2 = distance between z1 & z 2 .

z1 − z 2

z

2



z2

z1 z 1

Let C be a circle with centre z0 and radius ρ. Then such a circle C can be represented by C:z-z0= ρ . c z0

z-z0= ρ

Consequently, the inequality z-z0 < ρ ----------(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< ρ} is called a nbd. of z0

Deleted neighborhood: N0 = {z: 0 < z-z0< ρ } is called deleted nbd. It consists of all points z in an ρ -nbd of z0, except for the point z0 itself.

• The inequality z-z0>ρ 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.

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.

Domain: An open connected set is called a domain.

Accumulation point: A point z0 is said to be an accumulation point of a set S if every nbd N(z0) of z0 contains at least one point of S other than z0, i. e. if S∩ {N(z0)\{z0}} ≠ φ , then z0 is called an accumulation point of S. Remark: z0 may be or may not be a point

Ex1: Sketch & determine which are domains • 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 (− 2 ,0) &

radius 2. •S

is

a

unbounded

domain

and

it

is

Ex. 3

 z +1  S = z : < 1  z −1 

Sol. Note that : z + 1 < z - 1 2

⇒ z + 1 < z -1

2

⇒ (z + 1)(z + 1) < (z - 1)(z - 1) ⇒ x < 0. S is a domain and it is unbounded.