**for**

**Chapter 10 Conformal
Mapping**

**Overview**

The terminology "__conformal
mapping__" should have a familiar sound. In
1569 the Flemish cartographer Gerardus Mercator (1512--1594) devised
a cylindrical map projection that preserves angles. The
Mercator projection is still used today for world
maps. Another map projection known to the ancient Greeks
is the stereographic projection. It is also conformal (i.e., angle
preserving), and we introduced it in Section
2.5 when we defined the Riemann sphere. In complex analysis a
function preserves angles if and only if it is analytic or
anti-analytic (i.e., the conjugate of an analytic
function). A significant result, known as Riemann mapping
theorem, states that any simply connected domain (other than the
entire complex plane) can be mapped conformally onto the unit
disk.

**10.1 Basic Properties of
Conformal Mappings**

Let f(z)
be an analytic function in the domain D,
and let be
a point in D. If , then
we can express f(z) in the
form

(10-1) ,

where . If
z is near ,
then the transformation has
the linear approximation

,

where . Because when , for
points near
the transformation has
an effect much like the linear mapping . The
effect of the linear mapping S is a rotation of
the plane through the angle , followed
by a magnification by the factor , followed
by a rigid translation by the vector . Consequently,
the mapping preserves
angles at the point . We
now show that the mapping also
preserves angles at .

For a smooth curve that passes through the
point , we
use the notation

,

for .

A vector
tangent to C at the point
is given by

,

where the complex number is
expressed as a vector.

The angle of inclination
of with
respect to the positive x axis
is

.

The image of C under the mapping
is the curve K in the w
plane given by the formula

.

We can use the chain rule to show that a vector tangent
to K at the
point is
given by

.

The angle of inclination of
with respect to the positive u axis is

,

where .

Therefore the effect of the transformation is to rotate the angle of inclination of the tangent vector at through the angle to obtain the angle of inclination of the tangent vector at . This situation is illustrated in Figure 10.1.

The tangents at the points , where f(z) is an analytic function and .Figure 10.1

A mapping is
said to be angle preserving, or __conformal__
at ,
if it preserves angles between oriented curves in magnitude as well
as in orientation. Theorem 10.1 shows where a mapping by
an analytic function is conformal.

**Theorem 10.1 (**__Conformal
Mapping__**).** Let
f(z) be an analytic function in the
domain D, and let
be a point in D. If , then
f(z) is conformal at .

The analytic mapping is conformal at the point , where .Figure 10.2

**Example 10.1.** Show
that the mapping is
conformal at the points ,
,
and ,
and determine the angle of rotation given by
at the given points.

Solution. Because , we conclude that the mapping is conformal at all points except , where n is an integer.

Calculation reveals that

Therefore the angle of rotation is given by

Let f(z) be a nonconstant analytic function. If , then is called a critical point of f(z), and the mapping is not conformal at . The next result shows what happens at a critical point.

**Theorem 10.2.** Let
f(z) be analytic at the point . If
and , then
the mapping
magnifies angles at the vertex
by the factor k, as shown in Figure
10.3.

The analytic mapping at point , where and .Figure 10.3

**Example 10.2.** Show
that the mapping
maps the unit square
onto the region in the upper half-plane ,
which lies under the parabolas

and

as shown in Figure 10.4.

** Figure
10.4** The mapping .

Solution. The derivative is , and we conclude that the mapping is conformal for all . Note that the right angles at the vertices , , and are mapped onto right angles at the vertices , , and , respectively. At the point , we have and . Hence angles at the vertex are magnified by the factor . In particular, the right angle at is mapped onto the straight angle at .

Another property of a conformal
mapping is
obtained by considering the modulus of . If is
near , we
can use the equation

and neglect the term . We
then have the approximation

(10-9) .

From Equation (10-9), the distance between the images of the points and given approximately by . Therefore we say that the transformation changes small distances near by the scale factor . For example, the scale factor of the transformation near the point is .

We also need to say a few things about the
inverse transformation of
a conformal mapping near
a point ,
where .

A complete justification of the following assertions relies on
theorems studied in advanced calculus. (See, for instance, R.
Creighton Buck, Advanced Calculus, 3rd ed. (New York, McGraw-Hill),
pp. 358-361, 1978.)

We express the
mapping in
the coordinate form

(10-10) .

The mapping in Equations
(10-10) represents a transformation from
the xy plane into the uv plane, and the Jacobian determinant,
,
is defined by

(10-11) .

The transformation in Equations
(10-10) has a local inverse,
provided . Expanding
Equation (10-11) and using the
Cauchy--Riemann equations, we obtain

Consequently, Equations
(10-11) and
(10-11) imply that a local inverse
exists in a neighborhood of the point . The
derivative of g(w) at
is given by the familiar expression

**Exercises
for Section 10.1. Basic Properties of Conformal
Mappings**** **

__Mobius
- Bilinear Transformation__

**The Next Module
is**

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell