The Art and Craft of Problem Solving

4.2 COMPLEX NUMBERS 125

Let AB be a diameter oj a circle with diameter k. Consider a right

triangle ABC with right angle at B. Let D be the point oj intersection

(D of. A) oj line AC with the circle. Then

AD.AC = k^2.

(b) Consider the transformation J(z) = _

z

_. This is a function with complex do-

z - 1

main and range. Here is computer output (using Mathematica) of what J(z) does

to the domain (O:S Re (z) :s 2, - 2:S 1m (z) :s 2). The plot shows how J(z) trans­

forms the rectangular grid lines. Notice that J appears to transform the rectan­

gular Cartesian grid of the Gaussian plane into circles, all tangent at 1 [although

the point 1 is not in the range, nor is a neighborhood about 1, since z has to be

very large in order for J(z) to be close to 1].

Verify this phenomenon explicitly, without calculation, for the imaginary axis.

Prove that the function J(z) = _

z

_ transforms the imaginary axis into a circle

z-1

with center at (�, 0) and radius �. Do this two ways.

1. Algebraically. Let a point on the imaginary axis be it, where t is any real

number. Find the real and imaginary parts of JUt); i.e, put J(it) into the

form x+ yi. Then show that (x-1/2)^2 + y^2 = 1/4.

2. Geometrically. First, show that J(z) is the composition of four mappings,

in the following order:

Z f---> - Z f--->-Z, zf--->z+ l, Z f---> -.

Z' Z