The graphing window shows a sum of two damped oscillations,

\[x = A e^{at}\cos(bt) + B e^{ct}\cos(dt) = \text{Re}(Ae^{zt} + Be^{wt})\]

where \(z = a \pm bi\) and \(w = c \pm di\) and \(A, a, b, B, c,\) and \(d\) are real.

The complex conjugate pair \(a \pm bi\) is shown as green diamonds, and the complex conjugate pair \(c \pm di\) is shown as red diamonds, in the complex plane at lower right.

Control the parameters \(A, a, b, B, c,\) and \(d\) using the sliders below, or (in the case of \(a, b, c, d\)) by dragging the appropriate diamond in the complex plane window.

If \(A = 0\) the \(a, b\) sliders and the green diamond are turned off.
If \(B = 0\) the \(c, d\) sliders and the red diamond are turned off.

Click on the [Envelope] key to display an envelope, indicating the rate of growth or decay of \(x(t)\).

© 2001 H. Hohn and H. Miller