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RESIDUE THEORAM

By- venika ganjir

Contents

Terms

Introduction to residues

Residue theoram

Calculation of residues

Examples

Application

Bibliography

4

The residue of a function is the coefficient

of the term

in the Laurent series expansion (the

coefficient b

1

0

z  z

4

0

4

3

0

3

2

0

2

0

1

3

3 0

2

0 1 0 2 0

z z

b

z z

b

z z

b

z z

b

f z a a z z a z z a z z

Introductio

n

Where b

1

is the residue of f ( z ) at z

0

What’s so great about the Residue?

The formula for the coefficients of the

Laurent series says that (for f ( z ) analytic

inside the annulus)

4

0

4

3

0

3

2

0

2

0

1

3

3 0

2

0 1 0 2 0

z z

b

z z

b

z z

b

z z

b

f z a a z z a z z a z z





C

n

n

C

n (^) n

f z z z dz

j

dz b

z z

f z

j

a

1

1 0

0

So,

f ( z ) dz 2 jb

C

C

0

z

We can use it to evaluate

integrals

Theoram

Let f ( z ) be an analytic function in a

closed curve C except at a finite

number of singular points within C,

then

 







k

i

z z

C

f z dz j f z

i 1

( ) 2  Res ( )

C

8

Proof of Residue Theorem

Enclose all the singular points

with little circles C

1

, C

1

,  C

k

f ( z ) is analytic in here

By Cauchy’s Integral Theorm for multiply

connected regions:

   

   

k

C C C C

f ( z ) dz f ( z ) dz f ( z ) dz f ( z ) dz

1 2



C

But the integrals around each of the small

circles is just the residue at each singular

point inside that circle, and so

 





k

i

z z

C

f z dz j f z

i 1

( ) 2  Res ( )

Formula for finding the residue for a pole of

any order

If f ( z ) has a pole of order m at z

0

, then the

Laurent series is

m

m

z z

b

z z

b

z z

b

f z a a z z

0

2

0

2

0

1

0 1 0

lim  ( ) ( )

Res ( )

0 ( 1 )

( 1 )

0 0

z z f z

dz

d

m

f z

m

m

m

z z z^ z





^ 

m

m

m m m m

b z z b

z z f z a z z a z z b z z



 

2

2 0

1

1 0

1

0 0 0 1 0

now differentiate (m-1) times and let z  z

0

to get:

( 1 )^01

( 1 )

lim ( ) ( ) ( 1 )!

0

z z f z m b

dz

d

m

m

m

z z







Example

s

Let

Then f(z) has two poles: z = -2, a pole of order

1,and z=3 is a pole of order 2.

The residue at z = -2 is given by

13

 

 

 

 

 

     

2 2 2 2

2

2 2

2

2 2

2

2 2

2

0

2 2

0

2

2

2

2

2

0

1 1 1

2

1 1 1

2

1

,

1 1 1

1

2

1 1

1 ( 1 )

( )

1 1 1 1

Res ( 0 ) Res ( )

.

( )

1

( )( )

1

Res ( ) lim

.

1

( )( )

1

Res ( 0 ) lim

| | 1 ,| | | | isin thecircle, isoutof the circle.

Wehave 3 simplepoles, 0 , 2 / 1 0 1 1.

( 2 / 1 )

1 1

2

1 1 1

1 1 / / 2

1 / / 2

, isrealand| | 1.

1 sin

sin

Example 3 :

a a

a

a a

a

i

ia

I

a a

a

a

a

i

a

a

z z z^ i

z z

z z

z

z z

f f z

z z z

z

z z z z z

z

f z z z

z z z z z z z

z

f z

z z z z z z

a

a

i

z z iz a z

dz

z z iz a

z

ia

dz

z az iz a

z

iz i

dz

a z z i

z z i

I

a a

a

d

I

z z

z

C C C

  



  

  

  



















  

 





























  





 











 



 

 











 





  

       

 





  





 











  

 

 



 



  



 

 



   



     





  



C

r= 1

z +

z

-

z 0

14

arctan.

1

Or

.

( )( )

1

2 Res ( ) 2 lim

on theupperhalf plane

1

1

2 Residues of

1

Example 4 :

2

2

2



  



 



 











 

  











 



 





 







x

x

dx

z i z i

i f i i z i

z

I i

x

dx

I

z i

 

 

 .

2

'

( ) ( )

1

2 Res ( ) 2 lim

on the upper half plane

1

2 Residues of

, 0.

Example 5 :

2 2 3

2

2 2 2

2 2 2

z ai z ai a

i f ia i z ia

z a

I i

a

x a

dx

I

z ai



 



 











 

  















 

Conclusion

So the Residue allows us to evaluate

integrals of analytic functions f ( z )

over closed curves C when f ( z ) has

one singular point inside C.

NOTE-

The residue of a function at a

removable singularity is zero.

Bibliography

Higher engineering mathematics-

B.S.Agrewal (page no. 662-674)

Advanced engineering

mathematics- H.K.Das (page no.

515-532)

Mathmatical physics by Arfken

Wikipedia