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TitleComplex Variables
TagsMathematical Analysis Mathematical Relations Integral Continuous Function Holomorphic Function
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Page 2

Chapter 6 Mathematics




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

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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

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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




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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

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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 (








)

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