## 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 *z ^{2}*, 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) = z ^{2}* 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) = z ^{2}* 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:

Start ZMap by clicking on the above applet button.

*The applet should appear, with two horizontal lines and their images visible*Check the

*V-lines*and uncheck the*H-lines*checkboxes.

*You now see several vertical lines and their images, while the horizontal lines are no longer visible.*Click on the

*Rectangle Options*button

*A dialog window will appear where you can adjust the parameters for your lines*Adjust the parameters so that the

*Horizontal Lines*go from*-10*to*10*, while the*Vertical Lines*go from*1*to*2*. Also set the*number of lines*for the vertical lines to*2*(it does not matter how many horizontal lines you enter). Then click on*Okay*(or*Apply*).

*You should now see that vertical lines are mapped to left-right parabolas, opening to the left*To zoom out in the range, click

*Range zoom out*twice.

*You should now see see the entire parabolas in the range. If you move the mouse over the**y*-intercept on the range you can see that the*y*-intercepts of the right-most parabola are*z = -8 i*and*z = +8 i*, approximately.To recenter (pan) the range,

*double-click*on the origin*(0,0)*of the coordinate system in the range.

*The coordinate system in the range is panned so that the origin is now in the center of the window. You can**undo*all zoom/pan operations by clicking the appropriate*undo*buttons or , or by*right-clicking*on domain or range.To determine where an individual point in the domain is mapped, hold down the

*CTRL*-key and*click once*on the point in the domain where the blue vertical line intersects the real (x) axis.

*You will see a yellow dot appear in the domain where you CTRL-clicked as well as the image of that point in the range. You can see that the point with imaginary part zero is mapped to a point also with imaginary part zero. You could now conjecture that our map takes the real axis onto the real axis - is that true?*

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 - t^{2}+ i 4t

But that is equivalent to

x = 4 - 1/16 y^{2}

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.