1
APPENDIX 1: COMPLEX NUMBERS AND PROBABILITY
A.
Complex numbers and wave interference
The number
i
satisfies
i
2
=
−
1. Let us write a general complex number in cartesian and polar forms as
z
=
x
+
iy
=
re
iθ
,
where
x
=
Re
(
z
) and
y
=
Im
(
z
) are real numbers, called the real and imaginary parts of
z
, respectively;
r
≥
0
is the modulus and
θ
the argument (or phase) of
z
. The complex conjugate of any complex expression is found by
replacing
i
→ −
i
in the expression, e.g.,
z
∗
=
x
−
iy
is the complex conjugate of
z
. Since by simple geometry in the
Argand plane
x
=
rcosθ
and
y
=
rsinθ,
we have
e
iθ
=
cosθ
+
isinθ
. By using this formula and its complex conjugate
we find the important results,
e
iθ
+
e
−
iθ
= 2
cosθ
and
e
iθ
−
e
−
iθ
= 2
isinθ
.
Modulus
: Recall

z

2
=
zz
∗
≥
0
.
In terms of the cartesian representation,

z

=

x
+
iy

=
radicalbig
x
2
+
y
2
≥
0
.
The modulus
of a product is the product of the moduli,

z
1
z
2

=

z
1

z
2

(check:

re
iθ

=

r

e
iθ

=
r
×
1
,
as it should.) The modulus
of a sum of complex numbers is important in understanding interference of waves, i.e., the principle of superposition
for linear waves:

z
1
+
z
2

2
= (
z
1
+
z
2
)(
z
∗
1
+
z
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Kennedy
 mechanics, Complex Numbers, Probability distribution

Click to edit the document details