Graphics Reference
In-Depth Information
APPENDIX E
Basic Complex Analysis
E.1
Basic Facts
This appendix summarizes those facts about the complex numbers
C
and complex
analysis that are needed in other parts of the topic. A standard general reference on
complex analysis is [Ahlf66], but [Need98] is also a topic that the author would
recommend.
We can write a complex number
z
Œ
C
in the form
(
)
z
=
r cos
q
+
i
sin
q
.
Definition.
This representation is called the
polar form
representation of the
complex number
z
and the function arg(
z
) defined by arg(
z
) =qis called the
argu-
ment
function.
Note that arg(
z
) is a multiple-valued function. Also,
(
)
=
()
+
()
arg
zz
arg
z
arg
z
.
12
1
2
Definition.
The
extended complex plane
is obtained by adding one extra point to
C
.
This point is denoted by • and called the
point at infinity
. (Topologically, the extended
complex plane is the one-point compactification of
C
. See Section 5.5.)
We identify the extended complex plane with the unit sphere
S
2
in
R
3
using the
stereographic projection.
Definition.
The sphere
S
2
along with the complex structure induced by the stereo-
graphic projection is called the
Riemann sphere
.
A point (
z
1
,
z
2
,...,
z
n
) Œ
C
n
Definition.
is called a
real point
if all of its coordinates
z
i
are real.
Finally, since
C
and
R
2
are the same sets and we have a notion of limit and con-
tinuity for maps defined on
R
2
, these notions carry over to maps f :
C
Æ
C
.
