This applet is a graphical demonstration of Euler's formula:
eix = cos x + i sin x
We can generalize this to complex numbers:
ez = ex (cos y + i sin y)
where z = x + i y. In the applet, z is shown as a green dot on the complex plane. ez is shown as a red dot. The unit circle is shown in gray. z goes from -π to π and back again, which causes ez to go around the unit circle.

ez can be evaluated using a power series; the terms of this series are shown in alternating shades of blue, starting with 1 (the blue line from the origin to 1), and then z (the light blue line from 1 to 1+z), and then z2/2, and so on.

You can select another value of ez by clicking anywhere with the mouse. If you want to select another value of z with the mouse, change the popup menu from "Mouse = Select exp(z)" to "Mouse = Select z".

Huge thanks to Bob Hanson and his team for help converting this applet to javascript.