Complex analysis can do that...it is possible to have the same answer (i.e. magnitude) mirrored in a plane. I can't recall using residue theorem for "the Laurent expansion of the ordinary generating function of Chebyshev polynomials", though.
Yeah this type of problem is mathochistic. It's done the same way an explicit expression for Bessel functions is calculated by defining them in terms of the Laurent expansion of a generating function. The Bessel function's generating function has an exponential term that has to be expanded as a power series so when the Laurent integral is done term by term an infinite series results for each Bessel function. The Chebyshev generating function is a linear term over a quadratic and the Laurent integral can be done once using standard residue tables. So you wind up with finite expressions, the Chebyshev polynomials.
I was a math major as an undergrad so this kind of exercise was routine and not really abstract at all. I was also an EE major years later and this problem came up in filter design so I got curious and tried the Bessel procedure from a first course in complex analysis and it worked with a little modification. I recently got interested in circuits again here looking at led lighting and this problem was in a stack of papers. But yeah every night for 30 years this problem has haunted my dreams and I wake up screaming.
That's rather remarkable a 'problem' from 30 years ago haunts you still.
And yes, there is undoubtedly another approach. But why were you solving that problem? It sounds like it is something completely abstract.
I've only run into such 'weird' methods for solving the Helmholtz equation or problems involving wave equations IIRC, and they never were that complex.
That's funny you mentioned that, because I was thinking back, and the only time Chebyshev had any prominence in my education, was in electronics for exactly that