Java - Java

zMap: Quick Guide

The ZMap applet lets you investigate mapping properties of user-defined complex functions. When you start the program you will see two coordinate systems: one on the left representing the domain and one on the right representing the range. You can enter a function defined in terms of z, such as z2, to investigate. Then you can specify curves in the domain (horizontal lines, vertical lines, radii, arcs, or circles) to see how your function transforms them.

If you click the above example, the function f(z) = z2 has been pre-selected, together with two horizontal lines through z = 1/2 i and z = i. You will see that both lines are mapped to "left-right" parabolas, as shown before.

Question: Into what curves does the complex function f(z) = z2 transform vertical lines such as the vertical line through z = 2?

Answer: Before we check the answer mathematically, we use ZMap to come up with a good guess:

Of course we should verify this mathematically: a vertical line through z = 2 is given by l(t) = 2 + i t. Thus:

f(z) = f(2 + i t) = 4 - t2 + i 4t

But that is equivalent to

x = 4 - 1/16 y2

which is, indeed, a left-right parabola, opening to the left, with vertex at x = 4, and y-intercepts at y = -8 and y = +8. Thus, our rigorous math confirms our guess.