countor integral

Shri S’ad Vidya Mandal Institute of Technology
Topic: APPLICATION OF CONTOUR INTEGRATION
GUJARAT TECHNOLOGICAL UNIVERSITY
Complex variables and numerical method
PREPARED BY :
1] Rana manthan - 170450119044
2] Shah jay - 170450119046
3] Shah rishabh - 170450119047
4] Shah sheril - 170450119048

CONTENT
• Defining Line Integrals in the Complex
Plane
• Equivalence Between Complex and Real
Line Integrals
• Review of Line Integral Evaluation
• Line Integral Example
• Application of counter integral

Defining Line Integrals in the Complex Plane
0a z
Nb z
1z
2z
1
3z
2
3
N
1Nz C
nz
n
x
y
  
 
 
  
1
1
1
1
1
1
0
lim
lim
n
n
n n n
z
N
N n n n
n
n n n
b
N
N
a
z
N
n n n
N
n
C z z
I f z z
N
z z z
I f z dz I
f z z













 
 



  

 





Define on between and
Consider the sums
Let the number of subdivisions
such that and define
(The result is independent of the
details of the path subdivision.)
3
 
b
a
I f z dz 

Equivalence Between Complex and Real Line
Integrals
       
   
 
 
   
 
 
0 0 0 0
N N N N
b b
a a
b x ,y b x ,y
a x ,y a x ,y
C C
I f z dz u x,y iv x,y dx idy
u x,y dx v x,y dy i v x,y dx u x,y dy
udx vdy i vdx udy
C
 
 
     
   
   
 
 
 
Note that
The line integralis equivalent to two line integrals on .complex real
4

Review of Line Integral Evaluation
   
   
0
0
f
C
t
t
f
u x,y dx v x,y dy
dx dy
u v dt
dt dt
C
C : x x t , y y t , t t t
t

 
 
 
   


A line integral written as
is really a shorthand for
where is some parameterization of :
t
t
-a a
t-a a
1t
2t 3t
1Nt C
nt
x
y
0t
f Nt t
 ( ), ( )x t y t
1t 2t 3t 1Nt nt… …0t f Nt t
t
2 2 2
2 2
0
2 2
0
cos sin 0 2
f
f
x y a
x a t, y a t, t
x t , y a t , t a, t a,
x t , y a t , t a, t a,
 
  
    
     
      
parameterizations of the circle
1) (
Examp
)
2)
l
and
e :
5
The path C goes
counterclockwise
around the circle.

Line Integral Example
1
cos sin 0
C
I dz
z
C : x a , y a ,   

   
Evaluate : where
  2 2 2 2 2 2
1 1 1 x iy x y x y
f z i i
z x iy x iy x iy x y x y a a
         
                           
Consider
x
y
C

a
z
 
 
2
2
x
u z
a
y
v z
a



Hence
1 2 2
( )
a
a
xx y
I i dx
a a

   
    
  

   
1 2
C C
C C
I udx vdy i vdx udy
u iv dx v iu dy
I I
   
    
 
 
 
0
2 2 2
0
( )y x
I i dy
y
a a
  
    
  

6
The red color denotes functional dependence.

Line Integral Example (cont.)
Consider
x
y
C

a
z
 
0
2 2 2
0
0
2
0
0
2 2 2 2
2 2
0
2 2
2
0
2 2 2
1
2
0 2 2
0
( )
( )0
2
2 sin
2 2
2
a
a
a
a
y x
I i dy
a a
i
x dy
a
i i
a y dy a y dy
a a
i
a y dy
a
y a yi a
a
y
y
a
y
    

   
  
    
  
 
    
 
       
   



 

2
i
a
2
a
2
1
sin
2
a
a
i


  
  
   
 
  
 
7

Line Integral Example (cont.)
Consider
x
y
C

a
z 1 2
2 2
I I I i i
    
      
   
I i
Hence
Note: By symmetry (compare z and –z), we
also have:
1
2
C
dz i
z

x
y
C

a
z
8

Line Integral Example
Consider
x
y
C

r
z z
cos sin 0 2
cos sin
n
C
i
i
z dz
C : x r , y r ,
z r i r re ,
dz rie d ,


   
 

   
  



 Evalua : wherete
 
 
2
0
2
11
0
n
n i i
C
nn i
z dz re rie d
ir e d i

 





 

 

 1
1
ni
n e
r
i
 

 
 
 
2
0
12
1
( 1)
1
0 11
2 11
ni
n
n
n
, ne
r
i, nn





 

 
  
  
This is a useful result and it is
used to prove the “residue
theorem”.
9
Note: For n = -1, we can use the result on slide 9 (or just evaluate the integral directly).

10
Application of counter integral
• Oceanography
• Geology
• Environmental science
• Statistics
• Electrostatics
• meterology

countor integral

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