INDERPRASTHA ENGINEERING COLLEGE, GHAZIABAD

Department of Applied Sciences and Humanities

Session 2023-24
ENGINEERING MATHEMATICS II (BAS 203)

UNIT 5 – Complex Variables- Integration

Complex Integration ,Cauchy-Integral theorem, Cauchy integral

formula, Taylor’s and Laurent’s series, singularities and its
classification, zeros of analytic function , Residues, Cauchy’s Residue
theorem and its application
SHORT NOTES
Contour intergration
Intergal along a path in which intial and final points coincide ,so the curve C is a closed curve , then this
integral is called contour integral and is denoted by ∮ ( )
Note : If f(z)=u(x,y)+iv(x,y) , then since dz=dx+idy,
We have
∮ ( ) ∫( )( )
= ∫( ) ∫( )
Important definition
(i)Simply connected Region
A connected region is said to be a simply connected if all the interior points of a closed curve C drawn in
the region D are the points of the region D.
(ii)Multi- connected Region
Multiple connected region is bounded by more than one curve . We can convert a multiple connected
region into a simply connected region one , by giving it one or more cuts .
(iii) Zeros of an analytic function
The value of z for which the analytic function f(z) become zero is said to be the zero of f(z) .
Eg. Zero of .

CAUCHY’S INTEGRALTHEOREM (Cauchy-Goursat Theorem)

Statement: If f(z) is an analytic function and f '(z) is continuous at each point within and on a simple
closed curve C, then
∮ ( )

CAUCHY’S INTEGRAL FORMULA:

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

( )
f(a) = ∮ dz.
CAUCHY’S INTERGAL FORMULA FOR THE DERIVATIVESOF AN ANALYTIC FUCTION:

If a function f(z) is analytic in a region D, then its derivatives at any point z=a of D is also analytic in D
and is given by
( )
f '(z)= ∮( )
dz
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Theorem:

If a function f(z) as an analytic in a domain D, then at any point z=a of D, f(z) has derivatives of all
orders ,all of which are again analytic function in D, their values are given by

( )
( ) ∮( dz
)
Where C is any closed contour in D surrounding the points z =a.

REPRESENTATION OF A FUNCTION BY POWER SERIES:

A series of the form ∑ or ∑ ( ) whose terms are variable is called a power series ,
where z is a complex variables and ,a are complex constants .

TAYLOR’S SERIES

If f(z) is an analytic inside a circle C with centre at a , then for all z inside C,
( ) ( )
f(z)=f(a)+(z-a)f '(a)+ f ''(z)+…. ( )+….

Cor.1. Putting z=a+h in (3) , we get

f(a+h)=f(a)+hf '(a)+ f ''(a)+…. ( )+….
Cor.2. If a=0, the series (3) becomes
f(z)=f(0) zf '(0)+ f ''(0)+…. ( )+….

LAURENT’S SERIES

If f(z) is analytic inside and on the boundary of the annular (ring shaped) region R bounded by two
concentric circles C1 and C2 of radii r1and r2 (r1< r2 ) respectively having centre at a, then for all z in R,

f(z)=∑ ( ) +∑ ( )

( )
Where ∮( dw ;n=0,1,2,…
)

( )
And ∮( dw ;n=0,1,2,…
)

SINGULARITY

A singularity of a function f(z) is a point at which the function ceases to be analytic.

ISOLATED AND NON- ISOLATED SINGULARITY

If z=a is a singularity of f(z) and if there is no other singularity within a small circle surrounding the
point z=a , and z=a is said to be isolated singularity of the function f(z), otherwise it is called non-
isolated.

TYPES OF SINGULARITY

f(z)=∑ ( ) +∑ ( ) where 0< |z-a|<R.

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The second term ∑ ( ) on the RHS is called the Principal Part of f(z) at the isolated
singularity z=a . Now there arises three possibalities:
(i)All ’s are zero implies no term in P.P (Removable singularity)
(ii) Infinite number of terms in P.P (Essential singularity)
(iii) Finite number of terms in P.P(Pole)

Note:1 The limit point of the zero of a function f(z) is an isolated essential singularity.
Note:2 The limit point of the poles of a function f(z) is a non- isolated essential singularity.

DETECTION OF SINGULARITY
Removable singularity: ( ) exists and is finite.
Pole : ( )= .
Essential singularity: ( ) does not exist.

DEFINITION OF THE RESIDUE AT A POLE

The coefficient of b1 in a Laurents series is called residue of f(z) at the pole z=a . It is denoted by symbol
Res.(z=a)= b1.

RESIDUE AT INFINITY

Residue of f(z) at z= is defined as - ∫ ( ) where the intergrationis taken round C in anti-

clockwise direction.

CAUCHY’S RESIDUE THEOREM

Let f(z) be one valued and analytic with in and on a closed contour C except at a finite number of poles
z1, z2 , z3…….,zn and let R1, R2, R3,…..Rn be respectively the residues of f(z) at these poles , then

∮ ( ) ( R1+ R2+R+…..+Rn)= (Sum of the residues at the poles within C)

METHODS OF FINDING OUT RESIDUES

(i)If f(z) has a simple pole at z=a , then

Res{f(z)}= ( ) ( ).

(ii) If f(z) Has a pole of order m at z=a , then

Res{f(z)}=( *( ) f(z)]z=a
)

(iii) Residue of f(z) at z=

= ( ( )).

CONTOUR INTEGRATION

The process of integration along a contour is called contour integration.

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Integrals of the type ∫ ( ) where ( ) is a Rational Function of
and

∫ ( ) ∮ ( ) over the curve C

Where C is the unit circle |z|=1.

( ) ( )
Integrals of the type ∫ ( )
, where f(x) and F(x) are polinomials in x such that ( )

and F(x) has no zeros on the real axis.

( ) ( )
∫ ( ) ( )
in the upper half plane].

Cauchy’s integral theorem and Cauchy’s integral formula

Q.1)Evaluate ∫ ( )dz along the path y=x. (2019)

Q.2) Evaluate ∮ around the circle |z-2|=4
Q.3) Evaluate ∮ dz ; C | | (2022)
Q.4) Evaluate ∮ ( )
dz ,where c is (i) |z|=2 (ii)|z-1|= (iii) |z|=2 (2022)
Q.5) Write the statement of generalized Cauchy ‘s integral formula for nth derivative of an analytic
function at the point z=z0 . (2016)

Q.6)Evaluate ∮ dz where C is circle,

(i) |z|=3/2 (2016) (ii) |z-2|=1 (iii) |z|=1/2.

Q.7) Use Cauchy’s integral formula to evaluate

∮ ( ) dz where C is the circle|z|=3. (2017)
Q.8) Show that
(i) ∮ ( )
= ;C=|z-i|=2 (2018)
Q.9) Using Cauchy’s –integral formula, evaluate ∮ ( dz, where C
)( )
is a rectangle with vertices at 3±i,-2±i. (2023)
Q.10) Evaluate by Cauchy’s integral formula
∮( ) ( )
dz where C is the circle |z|=3 (2016)

Q.11) Evaluate ∮ dz ,where C |z-1|= . (2020)

Q.12) Using Cauchy’s integral formula evaluate ∮ ( )
dz where C is circle |z|=8 .
Q.13) State Cauchy’s integral theorem. (2016, 2022)
Taylor’s and Laurent’s series
Q.1) Expand in the region (a) |z|>2 (2019) (b) 1<|z|<2 (2015,19)
Q.2) Expand the following function in a Laurent’s series about the point
z=0: f(z)= . (2022)
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Q.3) Find the Taylor’s and Laurent’s series which represent the function
when (2022)
( )( )
(i) |z|<2 (ii) 2<|z|<3 (iii) |z|>3
Q.4) Expand ( in the regions
)( )
(i) |z|<1 (ii) 1<|z|<3 (2016)
Q.5) Define the Laurent’s series expansion of a function. Expand f(z)= ez/(z-2) in the Laurent series about
the point z=2. (2022)
Q.6) Find the Laurent’s expansion of f(z) =( )( )( ) in the region 3<|z+1|<3. (2017)

Q.7) Expand f(Z) = in Laurent series valid for the regions:

(i) 0<|z|<3 (ii) |z|>3 (2016)

Q.8) Expand f(z)= in the following regions:

(i) 0<|z|<1 (ii) 1<|z|<2 (iii) |z|>2 (2023)

( )
Q.9) Expand f(z) = in a Laurent series about the point z=2 .

Zeros and Singularities and its Classification

Q.1) Discuss the singularity of ( )

at z=a and z= . (2022)

Q.2) Discuss the singularity of sin( ). (2022)

Q.3) Classify the singularity of f(z)= . (2023)

Poles of analytic functions Residues and Cauchy’s Residue thoerem

Q.1) Find the poles( with its order) and residue at each pole of the following function f(z) = ( )( )
.
(2017)

Q.2) Find residue at each pole of the function ( )( )

and hence using Cauchy residue theorem ,
evaluate integral ∮ ( dz , where C: |z|=1 .
)( )
(2022)

Q.3) Find residue of f(z)= at z=0 . (2019)

( )

Q.4) Find the residue of f(z)= cotz at its pole . (2017)

Contour Integration

Q.1) Evaluate by contour integration : ∫ ,where a>|b|. Hence or otherwise evaluate

∫ , 0<a<1 . (2017)

Q.2)Evaluate by contour integration : ∫ cos(n )d ;

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 n . (2022)

Q.3) Evaluate by contour integration : ∫ d . (2018)

Q.4) Evaluate by contour integration : ∫ dx ,(a>b>0)

( )( )

(2017)

Q.5) Using contour integration , evaluate the real integral ∫ ,a>0 .

(2023)