The poles of a rational continuous-time LTI system determine exponential modes that can appear in its natural response.

For a pole

$$p=\sigma+i\omega,$$

the associated mode contains

$$e^{pt}=e^{\sigma t}e^{i\omega t}.$$

The real part controls growth or decay. If $\sigma<0$, the mode decays exponentially; if $\sigma>0$, it grows.

For a causal rational continuous-time system, BIBO stability requires all poles to lie in the open left half of the $s$-plane:

$$\Re(p)<0.$$

Poles on the imaginary axis do not decay, while right-half-plane poles grow.

Pole location therefore connects the algebra of the transfer function to time-domain behavior: horizontal position in the $s$-plane controls decay or growth, while the imaginary part contributes oscillation frequency.