##### Document Text Contents

Page 2

Chapter 6 Mathematics

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CHAPTER 6

Complex Variables

Complex Variable

, where x & y are real numbers and they are called real and imaginary part of Z.

| | √

( ) .

/

Function of a Complex Variable

I wh ch p c p ‘w’

‘ ’ c D,

then , w = f (z)

Where, z = x + iy,

w = f (z) = u(x, y) + i v(x, y)

Limits and Continuity

Let w = f(z) be any function of z defined in a bounded or closed domain D, then,

( ) = , if for every real we can find real

Such that

| ( ) | for | |<

Basically it means: Single value for all values of z in the neighbourhood of z = with the

possible exception of z = itself

z

f(z)

)

u X

v Y

z - Plane

pPlanePl

ane

w - Plane

Page 7

Chapter 6 Mathematics

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D

D

C

V = ∫ +∫( ) [

( )

]

V = 6y.

+ (-3

+ 3y) + c

= 3 y + 3y + c

f(z) = u + iv = – 3x + 3x + 1 + i (3 y - +3y) + c i

= ( ) + 3x + 1 + 3yi + c i

= ( ) + 3(x + iy) +1 + c i

= + 3z + 1 +c i

Complex integration

Line integral = ∫ ( )

, C need not be closed path

Here, f(z) = integrand

curve C = path of integration

Contour integral = ∮ ( )

, if C is closed path

If f(z) = u(x, y) + i v(x, y) and dz = dx + i dy

∫ ( )

= ∫ ( )

∫ ( )

Theorem

f(z) is analytic in a simple connected domain then ∫ ( )

= f( ) ( )

Integration is independent of the path

Dependence on path

I “C p ” p h p h p h

(However analytic function in simple connected domain is independent of path.)

C ch ’ theorem

If f(z) is analytic in a simple connected domain D, then for every simple closed path C in D,

-------(A)

∮ ( )

= 0

Page 8

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

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Note

I h w , C ch ’ h (z) is analytic on a simple closed path C and everywhere

inside C (with no exception, not even a single point) then ∫ ( )

C ch ’ I

If f(z) is analytic within and on a closed curve and if a is any point within C, then

f(a) =

∫

( )

f ’( )

∫

( )

( )

f ”( )

∫

( )

( )

.

f n(a) =

∫

( )

( )

Note

complex analytic function has derivative of all order.

in real calculus if a real function is differentiated once, nothing follows, about the existence of

second or higher derivative

Example

Find the complex integral of ∮

(C : circle of radius 3 )

Solution

∮

2π , because is analytic over entire region

Example

∮

= ? ( C : circle of radius 3 )

Solution

∮

= 0

Example

If g(z) =

∮ ( )

= ?

where, C is circle shown as,

Here C : |z – 1| = 1

1 0 2

Page 14

Chapter 6 Mathematics

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Note

C p h R h C ch ’

Basic concepts of complex number if z = x + iy

In trigonometric from

C

c

S

c

(c I )

Modulus of complex number

|z| = r .√

Argument of complex Number

= tan 1 = .

/

In exponential form

, c +

= 1

[ ]

| | √c

= 1

|z| = r =√

If the any pole is outside the closed contour

|z| = a

Its residue at this pole is always zero.

x

r

z

y

r =√

|z| = a

Page 15

Chapter 6 Mathematics

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We always find the residue at the poles,

where poles are inside the closed contour,

and for any outside point residue is zero.

Cube root of unity

ω2 = (

√

)

Point to remember

1. ω

2. ω ω2 = 0

3. , ω, ω2

(1, 0)

ω (

√

)

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

th

Cross, 10

th

Main, Jayanagar 4

th

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CHAPTER 6

Complex Variables

Complex Variable

, where x & y are real numbers and they are called real and imaginary part of Z.

| | √

( ) .

/

Function of a Complex Variable

I wh ch p c p ‘w’

‘ ’ c D,

then , w = f (z)

Where, z = x + iy,

w = f (z) = u(x, y) + i v(x, y)

Limits and Continuity

Let w = f(z) be any function of z defined in a bounded or closed domain D, then,

( ) = , if for every real we can find real

Such that

| ( ) | for | |<

Basically it means: Single value for all values of z in the neighbourhood of z = with the

possible exception of z = itself

z

f(z)

)

u X

v Y

z - Plane

pPlanePl

ane

w - Plane

Page 7

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

th

Cross, 10

th

Main, Jayanagar 4

th

Block, Bangalore-11

: 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 178

D

D

C

V = ∫ +∫( ) [

( )

]

V = 6y.

+ (-3

+ 3y) + c

= 3 y + 3y + c

f(z) = u + iv = – 3x + 3x + 1 + i (3 y - +3y) + c i

= ( ) + 3x + 1 + 3yi + c i

= ( ) + 3(x + iy) +1 + c i

= + 3z + 1 +c i

Complex integration

Line integral = ∫ ( )

, C need not be closed path

Here, f(z) = integrand

curve C = path of integration

Contour integral = ∮ ( )

, if C is closed path

If f(z) = u(x, y) + i v(x, y) and dz = dx + i dy

∫ ( )

= ∫ ( )

∫ ( )

Theorem

f(z) is analytic in a simple connected domain then ∫ ( )

= f( ) ( )

Integration is independent of the path

Dependence on path

I “C p ” p h p h p h

(However analytic function in simple connected domain is independent of path.)

C ch ’ theorem

If f(z) is analytic in a simple connected domain D, then for every simple closed path C in D,

-------(A)

∮ ( )

= 0

Page 8

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

th

Cross, 10

th

Main, Jayanagar 4

th

Block, Bangalore-11

: 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 179

Note

I h w , C ch ’ h (z) is analytic on a simple closed path C and everywhere

inside C (with no exception, not even a single point) then ∫ ( )

C ch ’ I

If f(z) is analytic within and on a closed curve and if a is any point within C, then

f(a) =

∫

( )

f ’( )

∫

( )

( )

f ”( )

∫

( )

( )

.

f n(a) =

∫

( )

( )

Note

complex analytic function has derivative of all order.

in real calculus if a real function is differentiated once, nothing follows, about the existence of

second or higher derivative

Example

Find the complex integral of ∮

(C : circle of radius 3 )

Solution

∮

2π , because is analytic over entire region

Example

∮

= ? ( C : circle of radius 3 )

Solution

∮

= 0

Example

If g(z) =

∮ ( )

= ?

where, C is circle shown as,

Here C : |z – 1| = 1

1 0 2

Page 14

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

th

Cross, 10

th

Main, Jayanagar 4

th

Block, Bangalore-11

: 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 185

Note

C p h R h C ch ’

Basic concepts of complex number if z = x + iy

In trigonometric from

C

c

S

c

(c I )

Modulus of complex number

|z| = r .√

Argument of complex Number

= tan 1 = .

/

In exponential form

, c +

= 1

[ ]

| | √c

= 1

|z| = r =√

If the any pole is outside the closed contour

|z| = a

Its residue at this pole is always zero.

x

r

z

y

r =√

|z| = a

Page 15

Chapter 6 Mathematics

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30

th

Cross, 10

th

Main, Jayanagar 4

th

Block, Bangalore-11

: 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 186

We always find the residue at the poles,

where poles are inside the closed contour,

and for any outside point residue is zero.

Cube root of unity

ω2 = (

√

)

Point to remember

1. ω

2. ω ω2 = 0

3. , ω, ω2

(1, 0)

ω (

√

)