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# 6.5 Complex Functions

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