**for**

**2.5 The Reciprocal
Transformation **

The mapping is
called the reciprocal transformation and
maps the z-plane one-to-one and onto the w-plane except for the
point z=0, which has no image, and the
point w=0, which has no preimage or inverse
image. Use the exponential notation
in the w-plane. If , we
have

.

The geometric description of the reciprocal transformation is now evident. It is an inversion (that is, the modulus of () is the reciprocal of the modulus of z) followed by a reflection through the x axis. The ray , is mapped one-to-one and onto the ray . Points that lie inside the unit circle are mapped onto points that lie outside the unit circle and vice versa. The situation is illustrated in Figure 2.21.

** Figure
2.21** The reciprocal
transformation .

We can extend the system of complex numbers by joining to it an "ideal" point denoted by and called the point at infinity. This new set is called the extended complex plane. You will see shortly that the point has the property, loosely speaking, that iff .

An -neighborhood of the point at infinity is the set . The usual way to visualize the point at infinity is by using what we call the stereographic projection, which is attributed to Riemann. Let be a sphere of diameter 1 that is centered at the point in three-dimensional space where coordinates are specified by the triple of real numbers . Here the complex number will be associated with the point .

The point
on is
called the north pole of . If
we let z be a complex number and
consider the line segment L in
three-dimensional space that joins z
to the north pole , then
L intersects in exactly
one point . The
correspondence is
called the stereographic projection of the complex z
plane onto the Riemann
sphere .

A point of
unit modulus will correspond with . If
z has modulus greater than
1, then will
lie in the upper hemisphere where for points we
have . If
z has modulus less than 1,
then will
lie in the lower hemisphere where for points we
have . The
complex number
corresponds with the south pole, .

Now you can see that indeed iff iff . Hence corresponds with the "ideal" point at infinity. The situation is shown in Figure 2.22.

**Figure
2.22** The complex plane and the Riemann
sphere .

Let us reconsider the
mapping . Let
us assign the images and to
the points and ,
respectively. We now write the reciprocal transformation
as

Note that the transformation is
a one-to-one mapping of the extended complex z plane onto the
extended complex w plane. Further, f is a continuous
mapping from the extended z plane onto the extended w
plane. We leave the details to you.

**Extra
Example.** Investigate the limits
of
as .

**Explore
Solution for Extra Example.**

**Example 2.22.** Show
that the image of the right half plane under
the mapping is
the closed disk in
the w-plane.

Solution. Proceeding as we did in Example 2.7, we get the
inverse mapping of as . Then

which describes the disk . As
the reciprocal transformation is one-to-one, preimages of the points
in the disk will
lie in the right half-plane . Figure
2.23 illustrates this result.

** Figure
2.23** The image of under
the mapping .

**Remark.** Alas, there
is a fly in the ointment here. As our notation
indicates, and are
not equivalent. The former implies the latter, but not
conversely. That is,
makes sense when , whereas
does not. Yet Figure 2.23 seems to indicate that f
maps onto
the entire disk , including
the point . Actually,
it does not, because
has no preimage in the complex plane. The way out of this
dilemma is to use the complex point at infinity. It is
that quantity that gets mapped to the point ,
for as we have already indicated, the preimage of 0
under the "extended" mapping is
indeed .

**Example 2.23.** For
the transformation , find
the image of the portion of the right half
plane that
lies inside the closed disk .

Solution. Using the result of Example 2.22, we need only
find the image of the closed disk and
intersect it with the closed disk . To
begin, we note that

.

Because , we
have, as before,

which is an inequality that determines the set of points in the w
plane that lie on and outside the circle . Note
that we do not have to deal with the point at infinity this time, as
the last inequality is not satisfied when . When
we intersect this set with , we
get the crescent-shaped region shown in Figure 2.24.

** Figure
2.24** The image of the half disk under
is a crescent-shaped region.

To study images of "generalized circles,"
we consider the equation

,

where A, B, C, and D are real numbers. This equation
represents either a circle or a line, depending on whether ,
respectively. Transforming the equation to polar
coordinates gives

.

Using the polar coordinate form of the
reciprocal transformation, we can express the image of the curve in
the preceding equation as

,

which represents either a circle or a line, depending on whether
,
respectively. Therefore, we have shown that the reciprocal
transformation
carries the class of lines and circles onto itself.

**Example
2.24.** Consider the
mapping .

**(a)** Find the images of
the vertical lines x =
a. **(b)** Find
the images the horizontal lines y = b.

Solution. Taking into account the point at infinity, we
see that the image of the line x=0 is the line u=0; that
is, the y axis is mapped onto the v axis.

Similarly, the x axis is mapped onto the u axis. Again,
the inverse mapping is , so
if , the
vertical line is
mapped onto the set of (u,v) points satisfying . For
(u,v)~=(0,0), this outcome is equivalent to

,

which is the equation of a circle in the w plane with
center and
radius . The
point at infinity is mapped to (u,v)=(0,0).

Similarly, the horizontal line is
mapped onto the circle

which has center and
radius . Figure
2.25 illustrates the images of several lines.

** Figure
2.25** The images of horizontal and vertical
lines under the reciprocal transformation.

**
**You are now ready to study Section
10.2 Bilinear
Transformations - Mobius
Transformations.

**Exercises
for Section 2.5. The Reciprocal
Transformation**** **

__Mobius
- Bilinear Transformation__

**The Next Module
is**

**Differentiable
and Analytic Functions**

**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