Chap
.
17
Conformal Mapping
Overview
. This chapter consists of five sections. In Sec. 17.1 you see that the mapping given by an analytic
function
f
u
z
U
is conformal (anglepreserving in both magnitude and sense of angles between curves;
Theorem 1). Exceptions are points at which the derivative is zero. Example:
f
u
z
U
u
z
2
,
f
U
u
z
U
u
2
z
u
0 at
z
u
0,
where angles are doubled. Sections 17.2 and 17.3 deal with the mapping by an important class of functions
w
u
u
az
±
b
U
/
u
cz
±
d
U
with constant
a
,
b
,
c
,
d
. Section 17.4 shows mapping by complex trigonometric and
hyperbolic functions. The optional Sec. 17.5 concerns an ingenious idea of making multivalued complex
relations singlevalued on suitable domains of definition, called
Riemann surfaces
.
To fully understand specific mappings, make graphs or sketches.
Sec
.
17
.
1
Geometry of Analytic Functions
:
Conformal Mapping
Problem Set 17
.
1
.
Page 733
3
.
Mapping by
w
u
f
(
z
)
u
_
z
. For instance, the segment from
u
0, 0
U
to any
u
x
,
y
U
u
u
0, 0
U
makes the angle
u
u
arctan
u
y
/
x
U
with the
x
axis. Its image is the segment from
u
0, 0
U
to
u
x
,
U
y
U
, which makes the angle
u
²
u
arctan
u
U
y
/
x
U
u
U
arctan
u
y
/
x
U
u
U
u
. Note that
f
u
z
U
is not analytic. Make a sketch.
5
.
Rotation
. The image of
z
u
re
i
u
(
r
³
0) under the mapping
w
u
iz
is
w
u
iz
u
e
i
U
/2
re
i
u
u
re
i
u
²
where
u
²
u
u
±
U
/2,
so that this mapping is indeed a rotation about 0 through an angle
U
/2 in the positive sense (a
counterclockwise rotation) .
In the answer on p. A42 the points on a line
x
u
c
are
z
u
x
±
iy
u
c
±
iy
,
so that
w
u
iz
u
i
u
c
±
iy
U
u
U
y
±
ic
.
For instance,
z
u
x
u
c
on the real axis is mapped onto
w
u
ic
on the imaginary axis. Similarly, for
y
u
k
you have
z
u
x
±
ik
, hence
iz
u
U
k
±
ix
.
7
.
Angular regions
,
sectors
. The double inequality
U
U
/4
´
Arg
z
´
U
/4 determines the angular region
R
between the bisecting lines of the first and fourth quadrants. Since
w
u
z
3
triples angles at the origin, the
image of
R
is the complex plane without the angular region between the bisecting lines of the second and
third quadrants; this image of
R
is an angular region of angle 3
U
/2, in which the image of the given
region lies, restricted to 
w

´
1/8 since 
z

´
1/2.
z