> So no need to start by questioning my knowledge on the topic, just get straight to the point.
Undergrad physics, so you are obviously more versed in the field than I am.
But, speaking as someone with some small background in this, I would hope that an article on 'science.org' that mentions Gauss and Riemann would go into slightly more detail than i = sqrt(-1). Even a two-liner description of the real and imaginary plane would be an improvement and would possibly motivate people who knew very little about the area into going and researching. The entire article is about possible periodicity in prime numbers - why, then, omit one of the most important things about complex numbers and their relationship to periodic systems? Euler's formula is a beautiful thing, and I say that as a luddite.
And as for the harmonic analsys as "something in physics used to separate sounds and their notes" - I mean... that's like saying "Moby Dick" is a book about a whale. Yes, it's technically correct, but there is such a lost opportunity to describe just how all-encompassing Fourier Analysis is and how it naturally ties back to the complex numbers mentioned previously.
So, as demanded, here:
For inputs, the function takes complex numbers, which are two-dimensional numbers with one coordinate on the real number plane and the other on the so-called "imaginary" plane. Complex numbers are fundamental to the description of many periodic systems such as waves, cycles, orbits etc.
harmonic analysis, which is a discipline that originated as the study of the composition of sound but was extended by mathematicians like Taylor and Fourier into a broad system of numerical analysis that is widely used in everything from number theory to neuroscience.
It would have taken very little additional effort, but the results would be rather different - showing paths forward rather than walls saying "this is all that there is to this".