Stability of Differential Equations on the Argand Plane

The negative half-plane is the condition for the exponent, being inside the unit circle is a condition for the basis.

That is, in linear differential equations with constant coefficients one usually uses a trial solution $y(t)=ce^{\lambda t}$. This is then bounded for positive times if the real part of the exponent factor $\lambda$ is non-positive.

In other contexts like difference equations (linear recursion equations) with constant coefficients or numerical ODE integration methods, especially the linear multi-step methods, the trial solution takes the form $y_n=cq^n$. This is bounded for positive integer $n$ if the basis $q$ is inside or on the unit circle.

Both can be connected if the exponential function is sampled on an equidistant sequence $t_n=nh$. Then $e^{\lambda t_n}=(e^{\lambda h})^n$, and, as $h>0$, $|e^{\lambda h}|\le 1$ is equivalent to $Re(\lambda)\le 0$.