Behaviour of x(t) = c e^(at)

The CT complex exponential is $x(t) = Ce^{at}$, where $C$ and $a$ may be complex. Its behaviour has three cases.

1. Real exponential ($C$, $a$ real)

• $a > 0$: grows exponentially (e.g. chain reactions, unstable systems).
• $a < 0$: decays exponentially (e.g. RC discharge, damped systems).
• $a = 0$: constant, $x(t) = C$.

2. Periodic complex exponential ($a = j\omega_0$, purely imaginary)

$x(t) = Ce^{j\omega_0 t} = C(\cos\omega_0 t + j\sin\omega_0 t)$. Magnitude is constant; real and imaginary parts are sinusoids. It is periodic for every $\omega_0 \neq 0$, with $T_0 = 2\pi/|\omega_0|$.

3. General complex exponential ($a = r + j\omega_0$, $C = |C|e^{j\theta}$)

$$x(t) = |C|e^{rt}\cos(\omega_0 t+\theta) + j|C|e^{rt}\sin(\omega_0 t+\theta)$$

• $r = 0$: sinusoid of constant amplitude.
• $r < 0$: damped sinusoid (decaying envelope $|C|e^{rt}$), e.g. RLC circuit response.
• $r > 0$: growing sinusoid.

The envelope $\pm|C|e^{rt}$ bounds the oscillation: it shrinks for $r<0$ and expands for $r>0$.