> I know what that is, having done quite a bit of complex maths, but the kind that uses i, not the kind that uses j

Some things in life leave a lasting impression[1]. :eye_roll:

It sounds like what you're looking for is an explanation of root locus analysis[2].

In the simplest control case, a transfer function is nothing more than the expression of a continuous closed-loop LTI system's output Y(s) over its input X(s) in the Laplace domain, conveniently abstracted as its forward path G(s) and negative feedback path H(s).

From there, Routh-Hurwitz method[3] can be used to determine stability of the system.

...and I will continue to use j, thanks.

[1] https://youtu.be/1rqJl7Rs6ps?t=1828

[2] https://en.wikipedia.org/wiki/Root_locus

[3] https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stabilit...