The Complex Exponential
This Demonstration plots the complex exponential . In the starting orientation ("above"), increases from -5 on the left to 5 on the right. You are looking down on the Argand plane along the positive imaginary axis—turn on the "Argand plane" checkbox to see it. (Note: animations may be jerky with this option on.) The real and imaginary parts of are controlled by the real and imaginary sliders. In the starting position, , and the plot is a left-handed helix. The slider controls the position of the gray ball.
Once you are familiar with , play with the imaginary slider to see how it controls the frequency and handedness of the helix. Now return to the "above" viewpoint, set the imaginary slider as close as you can to 0, then play with the real slider. You see the familiar real exponential.
Set to about and play with the slider. This is a shrinking spiral. A dynamic system with this time evolution is spiraling in toward a stable fixed point. Set to . This is an expanding spiral, such as you might see in the vicinity of an unstable fixed point. Look at this from the right viewpoint.