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.)