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When one studies a function from
to
, one often gets information by
looking at the graph of , which is a subset of
. If we consider a
function
, the graph of is a subset of
, and
is a ``
4-dimensional" object which cannot be visualized. We will now discuss a method to
represent functions from
to
geometrically.

** 6.31**
**Example (.)**
Let

be defined by

. If

is a point in the circle

, then

where

is a direction, and

is a point in the
circle

with radius

.
Thus

maps circles of radius

about

into circles of radius

about

. Let

be a direction in

. If

is
on the ray from

passing through

, then

for some

so

, which is on the ray from

passing through

. Hence the ray
making an angle

with the positive real axis gets mapped by

to the ray
making an angle

with the positive

-axis.

The left part of the figure shows a network formed by semicircles of radius

and rays making angles

with the positive

-axis. The right part of the figure shows the network formed by
circles of radius

and rays making angles

with the positive

-axis.

maps each semicircle in the left part of the figure
to a circle in the right part, and

maps each ray in the left part to a ray in the
right part. Also

maps each curvilinear rectangle on the left to a curvilinear
rectangle on the right. Notice that

, and in general

, so if
we know how

maps points in the right half plane, we know how it maps points in
the left half plane. The function

maps the right half plane

onto

.

** 6.32**
**Definition (Image of a function.)**
Let

be sets, let

, and let

be a subset of

.
We
define

and we call

the

*image of under *. We call

the

*image of* .

** 6.33**
**Example (, continued)**
In the figure above

6.5, the right half of the figure
is the image of the left half under the function

.
The figure below

6.5, shows the image of a
cat-shaped set under

. The cat on
the left lies in the first quadrant, so its square lies in the first two quadrants.
The tip of the right ear is

, with length

, and with direction making an angle

with the positive real axis. The image of the right ear has length

and makes an angle

with the positive

-axis. You
should examine how the parts of the cat in each curvilinear rectangle on the left
part of the figure correspond to their images on the right part.

** 6.34**
**Exercise.**
Let

be the cat shown in the left part of the above figure. Sketch the
image of

under each of the functions

below:

- a)
- .
- b)
- .
- c)
- .

** 6.35**
**Exercise.**
Let

be the cat shown in the left part of the above figure. Sketch the
image of

under

, where

.

** 6.36**
**Exercise.**
Let

be a direction in

; i.e., let

. Show that

.

** 6.37**
**Example.**
Let

for all

. If

is in the circle
of radius

, then

for some direction

, and

, so

takes points in the
circle of radius

about

to points in the circle of radius

about

.

Let be a direction. If is in the ray from through , then
for some
, so
. We noted
earlier that is the reflection of about the real axis, so maps the
ray making angle with the positive real axis into the ray making angle
with the positive real axis.
Thus maps the network of circles and lines in the left half of the figure
into the network on the right half.

The circular arcs in the left half of the figure have radii

Let's see how maps the vertical line
. We know that
and maps points in the upper half plane to points in the
lower half plane. Points far from the origin get mapped to points near to the
origin. I claim that maps the line into the circle with center
and radius
.

Let
, so is the set of points
in the line . Then

since

for all

. Hence,

and

maps every point in

into

. Now I claim that every point in

(except for

) is equal to

for some

.

Since
, it will be sufficient to show that if
, then . I want to
show

Well, suppose

,
and let

where

. Then

, so

so (by definition of

)

** 6.38**
**Exercise.**
The argument above does not apply to the vertical line

. Let

. Where does the reciprocal function

map

?

** 6.39**
**Entertainment.**
Let

for all

. Show that

maps
horizontal lines

into circles that pass through the origin. Sketch
the images of the lines

and the lines

on one set of axes using a compass. If you've done this correctly,
the circles
should intersect at right angles.

** 6.40**
**Exercise.**
- a)
- Sketch the image of the network of lines and circular arcs shown below under
the function , where for all
.

- b)
- Cube the cat in the picture.

** 6.41**
**Note.**
De Moivre's formula

, was first
stated in this form by Euler

in 1749 ([

46, pp. 452-454]). Euler named the
formula after Abraham De Moivre (1667-1754) who never explicitly stated the formula,
but used its consequences several times ([

46, pp. 440-450]).

The method for finding th roots of complex numbers:

was introduced by Euler in 1749 [

46, pp.452-454].

The idea of illustrating functions from the plane to the plane by distorting cat
faces is due to Vladimir Arnold (1937-??), and the figures are sometimes called ``
Arnold Cats". Usually Arnold cats have black faces and white eyes and noses, as in
[3, pp.6-9].

** Next:** 7. Complex Sequences
** Up:** 6. The Complex Numbers
** Previous:** 6.4 Square Roots
** Index**