17 Conformal Mapping
We now go back to the “main road”, which leads from Chapter 12 (PDEs),
via Chapters 13, 14 & 17, to Chapter 18 (Potential Theory in 2D).
In this chapter, we interpret complex functions
f
:
C
→
C
as mappings
from
R
2
to
R
2
, and thus consider a geometric approach to complex anal
ysis.
This new approach gives new insight on holomorphic functions; its
importance is similar to the study of curves
{
(
x, f
(
x
))
∈
R
2

x
∈
R
}
for real
functions.
Conformal mapping will also yield a standard method for solving bound
ary value problems in twodimensional potantial theory, by transforming a
complicated region into a simple one.
17.1 Geometry of Holomorphic Functions: Conformal
Mapping
We interpret a complex function
f
as a mapping of its domain of definition
(in
C
≃
R
2
) onto its image (in
C
≃
R
2
):
w
=
f
(
z
) =
u
(
x, y
) +
iv
(
x, y
)
,
z
=
x
+
iy.
(979)
We shall refer to such a mapping as “the mapping
w
=
f
(
z
)”, and we refer to
the “
z
plane” and the “
w
plane” to distinguish between the spaces containing
the domain and image of
f
. We use cartesian coordinates (
x, y
) and (
u, v
) or
polar coordinates (
r, ϑ
) and (
R, ϕ
) to represent points in the
z
 and
w
planes,
respectively.
Example:
The mapping
w
=
z
2
. Using polar forms
z
=
re
iϑ
in the
z
plane
and
w
=
Re
iϕ
in the
w
plane, we have
w
=
z
2
=
r
2
e
2
iϑ
, and therefore
R
=
r
2
,
ϕ
= 2
ϑ
(cf. Section 13.2). Hence circles

z

=
r
=
r
0
in the
z
plane
are mapped onto circles

w

=
R
=
r
2
0
in the
w
plane and rays arg
z
=
ϑ
=
ϑ
0
in the
z
plane onto rays arg
w
=
ϕ
= 2
ϑ
0
in the
w
plane.
In Cartesian coordinates we have
z
=
x
+
iy
and
u
= Re
w
= Re(
z
2
) =
x
2

y
2
,
v
= Im
w
= Im(
z
2
) = 2
xy.
(980)
Hence vertical lines
x
=
c
= const
.
in the
z
plane are mapped onto
u
=
c
2

y
2
,
v
= 2
cy
. So we obtain
v
2
= 4
c
2
y
2
= 4
c
2
(
c
2

u
)
,
(981)
174
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which defines for each
c
∈
R
a parabola in the
w
plane that opens to the left.
Similarly, horizontal lines
y
=
k
= const
.
in the
z
plane are mapped onto
parabolas in the
w
plane,
v
2
= 4
k
2
(
k
2
+
u
)
,
(982)
which are parabolas opening to the right, for each
k
∈
R
.
Conformal Mapping
A mapping
w
=
f
(
z
) is called
conformal
if it pre
serves angles between oriented curves, in magnitude as well as in sense.
Theorem 34
The mapping
w
=
f
(
z
)
by a holomorphic function
f
is con
formal, except at critical points, that is, points at which the derivative of
f
vanishes.
Proof:
We consider a curve
C
=
{
γ
(
t
)

t
∈
[
a, b
]
}
in the
z
plane, with
γ
: [
a, b
]
→
C
(cf. Section 14.1). The tangent to
C
at
z
0
=
γ
(
t
0
)
∈
C
is given
by ˙
γ
(
t
0
)
∈
C
, which is to be understood as a vector in the
z
plane, attached
to
z
0
. The image of
C
under
f
is given by the curve
C
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 Spring '08
 Staff
 Math, Holomorphic function, Möbius transformation, Conformal map, linear fractional transformation, linear fractional transformations

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