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Proof.
See [Ahlf66].
One important property of analytic functions is that they preserve the angles and
their sense (or orientation or sign) between intersecting curves. More precisely, let f(
z
)
be a function that is analytic at a point
z
0
. Let a
1
(t) and a
2
(t) be two curves in the
complex plane with a
1
(0) =a
2
(0) =
z
0
. Assume that both curves have nonzero tangent
vectors at
z
0
. Define
()
=
(
()
)
( )
=
(
()
)
()
=
()
b
t
f
a
t
,
b
t
f
a
t
,
and
w
=
b
0
b
0
.
1
1
2
2
0
1
2
If f¢(
z
0
) π
0
, then one can show the following:
(1) The angle between the tangent vectors a
¢(0) and a¢(0) at
z
0
is the same as the
angle between the tangent vectors b¢(0) and b¢(0) at
w
0
.
(2) The
sense
(or orientation or sign) of the angles is preserved, that is, if the
curves a
1
(t) and a
2
(t) are not tangent at
z
0
, then, thinking of the complex plane
as
R
2
, the ordered bases (a
¢(0),a¢ (0)) and (b¢(0),b¢ (0)) induce the same orien-
tation on
R
2
.
See Figure E.1.
Definition.
A function f(
z
) satisfying (1) and (2) at a point
z
0
is said to be
conformal
at
z
0
. A
conformal map
is a map that is conformal at every point of its domain.
E.2.3. Theorem.
An analytic function f(
z
) is conformal at every point
z
0
where
f¢(
z
0
) π
0
.
Proof.
See [Ahlf66].
In fact, one can show a converse:
E.2.4. Theorem.
If a function f(
z
) is conformal on an open set, then it is analytic
on that set.
Proof.
See [Need98].
The main result about real power series carry over to complex ones. A series
•
Â
0
n
f
()
=
z
a z
(E.3)
n
n
=
a
2
(t)
b
2
(t)
q
a
1
(t)
q
w
0
z
0
f
b
1
(t)
Figure E.1.
A conformal map.
