Complex Numbers
Stephanie Golmon
Principia College
Laser Teaching Center, Stony Brook University
June 27, 2006
Vectors



*http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1a.html
Vectors have both a
magnitude and a
direction.
Magnitude = 20 mi
Direction = 60˚
(angle of rotation
from the east)
Trigonometry

The black vector is
the sum of the two
red vectors
D = 20 mi
20 Sin (60˚)
17.32 mi
60˚
20 Cos (60˚)
10 mi
The Unit Circle
1
1
2
0.5
,
3
2
R=1
3
2
60˚
-1
-0.5
1
2
-0.5
-1
0.5
1
Radians vs. Degrees

radian: an angle of
one radian on the
unit circle produces
an arc with arc
length 1.

2π radians = 360˚

Example:
60˚= π/3 radians
*http://mathworld.wolfram.com/Radian.html
*http://www.math2.org/math/graphs/unitcircle.gif
*http://members.aol.com/williamgunther/math/ref/unitcircle.gif
Cosine, Sine, and the Unit Circle
Cos(t)
1
1
0.5
-6
-4
-2
0.5
2
4
6
2
4
6
-0.5
-1
-1
-0.5
0.5
1
Sin(t)
1
0.5
-0.5
-6
-4
-2
-0.5
-1
-1
Imaginary Numbers
1
4
1
i
2 or
i or
2
i
Complex Numbers

Have both a real and imaginary part
Z = 5 + 3i
Real part

Imaginary part
General form:
z = x +iy
Complex Plane
Imaginary axis
Real axis
*http://www.uncwil.edu/courses/mat111hb/Izs/complex/cplane.gif
Vectors in the complex plane




z=x+iy
A point z=x+iy can
be seen as the sum
of two vectors
x=Cos(θ)
y=Sin(θ)
Z=Cos(θ) + i Sin(θ)
R=1
i Sin(θ)
θ
Cos(θ)
Euler’s Formula
i
e  Cos( )  iSin ( )


Describes any point on the unit circle
θ is measured counterclockwise from
the positive x, axis
Proof of Euler’s Formula
z
dz
d
Cos
Sin
i Cos
dz
d
i Sin
i Cos
i Sin
iz
iz
Proof of Euler’s Formula
dz
1
z
z
z
ln z
e
i
i
z
i
id
Cos
i
e
i Sin
Polar Coordinates



i
Of the form r e
r is the distance to
the point from the
origin, called the
modulus
θ is the angle, called
the argument
20 ei
r= 20
θ= π/3
3
Polar vs. Cartesian Coordinates


Any point in the complex plane can be
i
written in polar coordinates (r e )
or in Cartesian coordinates (x+iy)
how to convert between them:
i
re
x
y
r Cos
r Sin
r Cos
i r Sin
r
x2
Tan
1
y2
y
x
Multiplying Complex Numbers


(5+5i)(-3+3i)= ?
-30
i 4
(5 2 e
)(3
30 e
30
i3
2 e
4
)= ?
Dividing Complex Numbers


(5+5i)/(-3+3i)= ?
i 4
(5 2 e
)/(3
5
3
e
i
2
i3
2 e
5
3
i
4
)= ?
Roots of Complex Numbers
1
ei
for n
0
i
e
i
e
2n
2
i or
i
i
2n
e
for n
1 2
i
1
i
2
e
i3
e
1 2
2
1 2
i
The Most Beautiful Equation:
i
e
1
0
Describing Waves
f ( z , t )  A cos[ k ( z  vt )   ]
1
is the amplitude
A
k ( z  vt )  

0.5
is the phase
is the phase constant
2

k
 /k
-5
-2.5
A
2.5
5
10
12.5
v
-0.5
-1
7.5

*derivation from: Introduction to Electrodynamics, Third Edition. David J. Griffiths. Upper Saddle River, New Jersey: Prentice Hall, 1999.
Continued…
2
one period T 
kv
1 kv v
frequency
f  

T 2 
angular frequency   2f  kv
f ( z , t )  A cos( kz  t   )
in complex notation:
i kz
f ( z , t )  Re[ A e
t
]