Download Notes 1 6382 Introduction To Complex Variables

ECE 6382
Fall 2016
David R. Jackson
Notes 1
Introduction to Complex Variables
Notes are from D. R. Wilton, Dept. of ECE
1
Some Applications of Complex Variables
 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)
2
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.
z  x  iy
 i ( j ) 
 r  cos   i sin 
i
 re
 r 

1, x  Re z , y  Im z


z   x, y 
(from figure)
(Euler formula (not yet proved!))
(polar form)
 z  arg z (polar form)
y
r
z
z

x
r  z  x 2  y 2  magnitude of z
y
  arg z  tan
 argument or phase of z
x
x  r cos  , y  r sin 
1
Argand diagram
3
Complex Arithmetic and Algebra (cont.)
y
Addition / subtraction :
z1  z2   x1  iy1    x2  iy2 
  x1  x2   i  y1  y2 
z1 - z2
Multiplication :
z1 z2   x1  iy1  x2  iy2 
-z2
y1
y2
z1 + z2
z1
1
x1
2
z2
x2
x
  x1 x2  y1 y2   i  x1 y2  x2 y1 
 i   0  i1 0  i1   1 
2
Division :
1

  0,1 0,1   1, 0 
1
x1  iy1 x2  iy2

x2  iy2 x2  iy2
x x  y y  i ( y1 x2  y2 x1 )
 1 2 1 22
x2  y22
z1 / z2 

2
Division is kind of messy in rectangular coordinates!

 x x  y y y x  y2 x1 
z1 / z2   1 22 12 2 , 1 22

2
x

y
x

y
2
2
2
 2

4
Complex Arithmetic and Algebra (cont.)
y
z
z
r
x
r

z*
Conjugation :
z *   x  iy 
z1 z2*
z1 z2*
Note : z1 / z2 

2
*
z2 z2
z2
Magnitude :
r z 
x2  y 2  z z* 
 re  re 
i
 i
5
Euler’s Formula
Recall :

x2 x3
xn
e  1 x   
 
2! 3!
n 0 n !
Define extension to complex variable ( x  z  x  iy ) :
x

z2 z3
e  1 z   
2! 3!
zn
 
n 0 n !
z

 e 
i
n 0
 i 
n
n!
 1
2
2!

(converges for all z )
4
4!


3 5
 i    
3! 5!




 cos   i sin 
 ei  cos   i sin 
e  i  cos   i sin 
More generally,

eiz  cos z  i sin z
e  iz  cos z  i sin z
eiz  e  iz
cos z 
2
eiz  e  iz
sin z 
2i
6
Application to Trigonometric Identities
Many trigonometric identities follow from a simple application of Euler's formula :
ei 2  cos 2  i sin 2
On the other hand,
ei 2   ei    cos   i sin    cos 2   sin 2   i  2 cos  sin  
2
2
Equating real and imaginary parts of the two expressions yields two identities :
cos 2  cos 2   sin 2 
sin 2  2 cos  sin 
e 1
i   2 
 cos 1   2   i sin 1   2 
On the other hand,
e 1
i   2 
 ei1 e  i2
  cos 1  i sin 1  cos  2  i sin  2 
  cos 1 cos  2 sin 1 sin  2   i  sin 1 cos  2  cos 1 sin  2 
Equating real and imaginary parts yields :
cos 1   2   cos 1 cos  2 sin 1 sin  2
sin 1   2   sin 1 cos  2  cos 1 sin  2
7
Application to Trigonometric Identities (cont.)
In general,
e
in
 
 cos n  i sin n  e
i n

 cos   i sin  
n
Expand using binomial theorem,
then equate real and imaginary parts
to obtain new identities.
eiz  cos z  i sin z
eiz  cos z  i sin z


eiz  eiz
eiz  e iz
cos z 
, sin z 
2
2i
e z  e z
e z  e z
cos  iz  
 cosh z , sin  iz   
 i sinh z
2
2i
8
DeMoivre’s Theorem
 
z n  rei
n
 r n ein  r n  cos n  i sin n   r n n
(DeMoivre's Theorem)
Note that for n an integer, the result is independent of how  is measured
 rei  2 k    r n ei n  2 kn   r n cos n   2 kn  i sin n   2 kn  (k an integer)









 r n  cos n  i sin n 
n
 zn
y
z
r
,
k  1
z
 ,k 0
x
, k  1
9
Roots of a Complex Number
z n   rei   r n ein  r n  cos n  i sin n   r n n (DeMoivre's Theorem)
n
Applies also for n not an integer, but in this case, the result
is not independent of how  is measured.
Example : n th root of a complex number :
1
zn
  re
1
i   2 k   n
 



1 i n  2 kn
r ne


1
rn


e.g.,  8i 

cos   2 k  i sin   2 k  , k  0,1, 2,
n
n
n
n 

1
1
3

n roots
i  i 2 k
 i 2 i 2 k  3
6
3  2 cos   
  8e

2
e

  6
 


 2 k   i sin     2 k   ,
3 
 6
3 
n 1
k  0,1, 2
 3
  
1
  
 2 cos     i sin      2 cos  30   i sin  30    2 
i  
2
 6 
  6
 2
   2 
  2  
 2 cos   

  i sin   
   2 cos  90   i sin  90  
6
3
6
3



 
   4 
  4  
 2 cos   

i
sin

 
 
6
3
6
3



 
2 cos  210   i sin  210  
3  i,
2i,
  3  i,
10
Roots of a Complex Number (cont.)
w  z1/3   8i 
1/3
y
z
 8i 
1
3
v
w
w  u  iv
 3  i,

  2i,

 3  i
x
u
8i
Im
1120
Note that the n th root of z can also be expressed in terms
of the n th root of unity :
1
n
z
  re

where
1
i  2 k   n
1
 

1
n
n th root
of unity

 e
i 2 k

1 i n 2 nk
n
r e

1 i
n
r en
e
i 2 kn
1  0
Re
1 240
Cube root
of unity
"principal" n th root
of unity
root of z

1
n
e
i 2 kn
 cos
2 k
2 k
 i sin
,
n
n
k  0,1,
,n 1
11