Complex Angles

What happens if we take the exponential of a quantity that is neither pure real nor pure imaginary, but a little of both? We can do this several ways, but in view of our interest in waves I will put it this way: suppose that instead of $\theta$ we have an argument $z = (\omega + i \lambda)t$ where $\omega$, $\lambda$ and t are all real. Then

$$e^{iz} \; = \; e^{i(\omega + i \lambda)t} \; = \; e^{i\omega t} \cdot e^{-\lambda t}$$

That is, we have an oscillatory function multiplied by an exponentially decaying "envelope" function -- the phenomenon of DAMPED OSCILLATIONS that describes virtually every actual case of oscillatory motion.