Introduction to Complex Variables
Dr.M.Sheela
Assistant Professor of Mathematics
1
ComPlex Analysis

Some Applications of Complex Variables
2
 Phasor-domain analysis in physics and engineering
 Laplace and Fourier transforms
 Evaluation of integrals
 Asymptotics (method of steepest descent)
 Conformal Mapping (solution of Laplace’s equation)
 Radiation physics (branch cuts, poles)

A complex number z may be thought of simply as an ordered pair of real
numbers (x, y) with rules for addition, multiplication, etc.
   
 
2 2
1
( ) 1, Re , Im
cos sin
arg
ar
,
g tan
i
z x iy i j x z y z
r i
z
re
r
z z
r z x y
x
z
z
x
y
y

 

 
      
 

 
 
   
 
   
 
 
(from figure)
(Euler formula (not yet proven!))
(angle notation)
(angle notation)
magnitude of
argument or phase
cos , sin
z
x r y r  
of
z
x
y

r
Argand diagram
(polar form)
z
3
Note: In Euler's formula, the
angle  must be in radians.
Note: Usually we will use i to denote the square- root of -1.
However, we will often switch to using j when we are doing an engineering example.

   
   
1 2 1 1 2 2
1 2 1 2
z z x iy x iy
x x i y y
    
   
Addition / subtraction:
4
Geometrically, this works the same way and adding and subtracting two-
dimensional vectors:

  
   
  
    1 21 2
2 2
2 2
1 2 1 2 1
1
1 2 1 1 2 2
1 2 1 2 1 2 2 1
2
2
1 2 1 2 1 2
1 1
1 2
2 2
1 2 1 2 1 2 2 1
2 2
22 2
2 2
2
1
0 1 0 1 1
/
/
(
,
)
ii i
z z x iy x iy
y
x x y y i x y x y
i i i
z z re r e rr
x iy
x iy
x x yx y
e
x iy
z z
x iy
x x y y i y y x
x y
z z
x y
   

  
   
     

 





 



 


64 7 48
Multiplication:
Division:
     1 21 2
2 2 1
2 2
1
1 2 1 2
2
2 2
/ /
ii i r
z z re
x
r e e
r
y x
x y
    

 
 
 






5
Multiplication and division are easier in polar form

6
 We can multiply and divide complex numbers. We cannot divide two-
dimensional vectors.
 We can, however, multiply two-dimensional vectors in two different ways
(dot product and cross product).
Important points:

 
  
*
* *
1 2 1 2
1 2 2*
2 2 2
2 2 *
/
i i
z x iy
z z z z
z z
z z z
r z x y z z re re 
 
 
    
Note :
Conjugation:
Magnitude :
z
x
y

r
z

r
z*
7

Euler’s Formula
 
2 3
0
2 3
0
2 4 3 5
0
1
2! 3! !
1
2! 3! !
1
! 2! 4! 3! 5!
cos sin
n
x
n
n
z
n
n
i
n
i
x x x
e x
n
x z x iy
z z z
e z
n
i
e i
n
i
e
z


    

 






     
  
     
 
          
 
 




L
L
L L
Recall:
Define extension to a complex variable ( ):
(converges for all )
cos sin cos sin
cos sin cos sin
cos sin
2 2
i
iz iz
iz iz iz iz
i e i
e z i z e z i z
e e e e
z z
i

   

 



  
  
 
 
More generally,
8
   cos cosh , sin sinh
2 2 2
z z z z z z
e e e e e e
iz z iz i i z
i
  
  
     

Application to Trigonometric Identities
     
2
2 22 2 2
cos2 sin 2
cos sin cos sin 2cos sin
i
i i
e i
e e i i

 
 
     
 
     
W
Many trigonometric identities follow from a simple application of Euler's formula :
On the other hand,
Equatingreal andimaginary parts of t
 
   
 
  
 
1 2
1 2 1 2
2 2
1 2 1 2
1 1 2 2
1 2 1 2 1 2 1
cos2 cos sin
sin 2 2cos sin
cos sin
cos sin cos sin
cos cos sin sin sin cos cos
i
i i i
e i
e e e
i i
i
 
   
  
  
   
   
      

 
 

   

  
  
W
m
he two expressions yields identities:
On the other hand,
two
 
 
 
2
1 2 1 2 1 2
1 2 1 2 1 2
sin
cos cos cos sin sin
sin sin cos cos sin

     
     
 
  
m
Equatingreal andimaginary parts yields :
9

DeMoivre’s Theorem
10

2 k 
2 k 
x
y
z
z
   
   
   
2 2
cos sin
cos 2 sin 2
co
nn i n in n
n
i k i n knn n
n
z re r e r n i n
n
re r e r n kn i n kn k
r
 
   
 

   
 
   
           

W
W
(DeMoivre's Theorem)
Note that for aninteger, the result is of how is measured
( aninteger)
independent
 s sin
n
n i n
z
 


Roots of a Complex Number
   cos sin
nn i n in n
th
z re r e r n i n
n
n
 
 

   W
W
(DeMoivre's Theorem)
Applies also for not aninteger,but in this case, the result
is independent of how is measu
m
no red.
root ofp a cx om: pl
t
ea xE m l nue
   
   
 
22
22
6 32
1
1 1 1
1
31
3
2 2cos sin , 0,1,2, 1
2 2
8 8 2 2 cos sin , 0,1,2
6 3 6 3
2 cos
6
k
n nii k
ki ii i k
n
n
n n n k k
n n n nz re r e r i k n
k k
i e e i k
  
  
   
   


  
           
      
             
     

 


 
 
6 44 7 4 48
L
roots
ber :
e.g.,
   sin 2 cos 30 sin 30 2
6
i i
   
               
   
3
2
1
2
i
   
   
3 ,
2 2
2 cos sin 2 cos 90 sin 90 2 ,
6 3 6 3
4 4
2 cos sin 2 cos 210 sin 210 3 ,
6 3 6 3
i
i i i
i i i
   
   
 
  
 
    
                 
    
    
                   
    
11

Roots of a Complex Number (cont.)
 
   
1
3
222
1
1 1 1
3 ,
8 2 ,
3
k
n
k
n n n
ik
z
ii i
n
n
n n n
i
i i
i
n z
e
n
z re r e r e
   
 

  

 
    
 
 1 2 3
W
"principal" throot
of unityroot of
Note that the throot of can also be expressedin terms
of the :th root of unity
 {  
11 22 2 2
1 cos sin , 0,1, , 1n
k
nn ii k
n
k k
e ie k n
n n
   
     
1 2 3
L
throot
of unity
where
z
x
y
8i
u
v
w
 
1/31/3
8w z i  
Re
Im
1 0 
1 120 
1 240 
Cube root
of unity
12
w u iv 
Example (cont.)