1.
Notes are adapted from D. R. Wilton, Dept. of ECE
ECE 6382
Introduction to Complex Variables
David R. Jackson
1
Fall 2022
Notes 1

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

3.
Complex Arithmetic and Algebra
A complex number z may be thought of simply as an ordered pair
of real numbers (x, y) with rules for addition, multiplication, etc.
   
 
R ,
( ) 1, e , Im
cos sin
i
z x iy i j y z x
x z z
r i
y
re
 
      
 

 
(from figure)
(Euler formula (not yet proven!))
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.
Argand diagram
(polar form)
Note:
We can say that
1
i  
But we need to be careful to properly interpret
the square root (using the principal branch). This
is what the radical sign usually denotes.
  
  
x
y
r

z
z plane
 
arg
z r
z 



4.
Complex Arithmetic and Algebra
4
x
y
r

z
z plane
Note on phase angle (argument):
The phase angle  is non-unique. We
can add any multiple of 2 (360o) to it.
This does not change x and y.
Principal branch:
  
  
The most common choice
for the “principal branch” is*:
Note:
Adding multiples of 2 to  will affect
some functions, but not others.
Examples:    
f z z
 noeffect
   
1/2
f z z
 willeffect
2
p n
  
 
p
  
  
*e.g., the one that Matlab uses

5.
Complex Arithmetic and Algebra (cont.)
   
   
1 2 1 1 2 2
1 2 1 2
z z x iy x iy
x x i y y
    
   
Addition / subtraction:
5
Geometrically, this works the same way and adding and subtracting two-dimensional vectors:
“tip-to-tail rule”
x
y
1
z
2
z
1 2
z z

x
y
1
z
2
z
1 2
z z

2
z


6.
Complex Arithmetic and Algebra (cont.)
  
   
  
    
1 2
1 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
2
2 2
2 2
2
1
0 1 0 1 1
/
/
(
,
)
i
i 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 y
x y
e
x iy
z z
x iy
x x y y i y y x
x y
z z
x y
 
  

  
   
     

 





 



 


Multiplication:
Division:
     
1 2
1 2
2 2 1
2 2
1
1 2 1 2
2
2 2
/ /
i
i i r
z z re
x
r e e
r
y x
x y
 
  
 

 

 
 






6
Multiplication and division are easier in polar form!
1/ i i
 
Example :

7.
Complex Arithmetic and Algebra (cont.)
7
 We can multiply and divide complex numbers. We cannot
divide two-dimensional vectors.
Important point:
(We can, however, multiply two-dimensional vectors in two different
ways, using the dot product and the cross product.)

8.
Complex Arithmetic and Algebra (cont.)
 
  
  
*
*
2 2 *
*
i i
z x iy
z z z
z r x y x iy x iy z z
z r re re z z
 

 

      
  
To see this :
Conjugation:
Magnitude :
8



y
x
r
z
r
*
z

9.
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


    

 






     
  
     
 
         
 
 
 




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,
9
   
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
  
  
     

Note: The variable  here
is usually taken to be real,
but it does not have to be.
Leonhard Euler

10.
Application to Trigonometric Identities
     
2
2 2
2 2 2
cos2 sin 2
cos sin cos sin 2cos sin
i
i i
e i
e e i i

 
 
     
 
     
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
 
   
  
  
   
   
      

 
 

   

  
  
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

     
     
 
  
Equatingreal andimaginary parts yields:
10

11.
DeMoivre’s Theorem
11

2
 

2
 

x
y
z
z
   
   
   
2 2
cos sin
cos 2 sin 2
p p
p p
n
n i n in n
n
i k i n kn
n n
z re r e r n i n
n
re r e r n kn i n kn k
 
   
 

   
 
   
   
    
   
 

(DeMoivre's Theorem)
Note that for aninteger, the result is of how is measured
( anint ge
e r)
independent
 
cos sin
p p
n
n
r n i n
z
 


Abraham de Moivre

12.
Roots of a Complex Number
 
   
2
2
1
1 1 1
2 2
cos sin , 0,1,2, 1
p
p p
k
n n
i
i
n
k n
n n n k k
n n n n
z re r e r i k n


   
 
 
 
 


   
    
   
   
  
roots
12
1
n
w z

     
1
3
0: 8 2 cos sin 2 cos 30 sin 30 2
6 6
k i i i
 
 
   
            
 
   
   
   
 
3
2
1
2
i

     
     
1
3
1
3
3 ,
2 2
1: 8 2 cos sin 2 cos 90 sin 90 2 ,
6 3 6 3
4 4
2 : 8 2 cos sin 2 cos 210 sin 210 3 ,
6 3 6 3
i
k i i i i
k i i i i
   
   
 
 
 
 
 
   
            
 
   
   
   
 
 
   
              
 
   
   
   
 
 
2
2
6 3
2
1
3
1
3
2 2
8 8 2 2 cos sin , 0,1,2
6 3 6 3
k
i i
i i k k k
i e e i k
 
     
 
 
   
   
       
     
 
   
 
 
 
E l
xamp e:
 In this case the results depend on how  is measured.

13.
Roots of a Complex Number (cont.)
 
 
1
3
2
2 2
1
1 1 1
3 ,
8 2 ,
3
p
p
p
k
n n k
n
n
i
i k i
n
i
n
n n n
i
i i
i
n
n
z re r e r e
e
z



  
 
 
 


 

  

 

 
 
 
 

"principal throot
o
branch"
Note that the throot of can also be expressedin terms
of the :
th root of unity
   
1
1 2
2 2 2
1 cos sin , 0,1, , 1
k
n
n
n i k
n
i k k
e
e i k n
n n
   
     
f unity
throot
of unity
where
z
x
y
8i

u
v
w
 
1/3
1/3
8
w z i
  
Re
Im
1 0
 
1 120
 
1 240
 
Cube root
of unity
(n = 3)
13
w u iv
 
Example (cont.)