I'm currently an undergraduate electrical engineering student who recently took an elective mathematical course on complex analysis, where we learned about all kinds of singularities for complex numbers, like poles.
This semester, in a circuit analysis course, we are finding the poles of transfer functions, but the concepts don't make sense in my head.
Using an easy example (from Wikipedia), say we have a transfer function of
\$H(s) = \frac{1}{1 + RCs}\$, where they say that the pole is at \$s = -\frac{1}{RC}\$, which, sure, works out at the moment. However, \$s = j\omega\$, so equating the two,
\$-\frac{1}{RC} = j\omega\$
\$\frac{j}{RC} = \omega\$
But now this means that our angular frequency, \$\omega\$, is imaginary, which I thought was not possible due to \$\omega\$ being a real number, as the frequency of a circuit would only be real (please let me know if my assumption is wrong).
So basically, I don't understand how there can possibly be a pole for that transfer function, as we have a constant non-zero real part on the denominator, accompanied by a variable imaginary term, \$j\omega\$. Given that I am assuming that \$\omega\$ can only ever be real, then the denominator can never approach 0, thus not making it a pole.
Have I made an incorrect assumption somewhere, is my understanding flawed, or is there a mismatch between mathematics and electrical engineering in what a "pole" is?