I have had two professors, from different schools, both eminent in their respective fields, openly telling the classes I enrolled in that a solid background in complex variable/analysis is probably the most important thing to learn in order to do control theory well and all of frequency-domain control theory is an application of these disciplines.

At the same time, I found this topic to be the most easily forgettable because if you don't work with it (and nowadays few do in my experience), you forget it easily. And I don't mean what is a complex number. I mean things like the Cauchy contour integral, Argument principle and other things that are helpful to derive certain results for linear discrete-time systems and drawing loops around certain numbers. I am pretty sure none of my classmates remember any of this stuff 1 year out of university.

I could never quite get over between this seeming mismatch between the lack of complex variable/analysis I see in practical engineering (or advanced things in control) and what they said. Both of them were quite old too so maybe this is some kind of retrospective wisdom that young people wouldn't get. Or they could just be out of date. Or it could be some kind of over-claiming to get people to pay attention.

Is all of frequency-domain control fundamentally an application of complex variable/analysis? Is everything in frequency-domain control fundamentally linked to some kind of result in those math areas?