# Applications of complex variables beyond undergrad syllabus

So complex numbers solve all polynomials, appear as eigenvalues, appear in intermediate calculations in solving cubics, relate trig to hyperbolic functions, can be used to contour integrate real functions more easily, can represent fourier series more compactly, describe calculations about wave phenomena, used in potential theory and conformal maps.

But what are some unusual, not well known or just advanced applications of complex numbers that one would be unlikely to encounter in an undergraduate mathematics degree ?

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You could flick through this book on special functions to see some applications of contour integration to prove certain identities and other useful stuff like asymptotic estimates. –  Sam Jul 13 '12 at 22:05

Cauchy's Theorem is a staple in analytic combinatorics and analytic number theory. An example accessible to undergraduates could be finding the exact formula for the Fibonacci numbers by applying the residue theorem to the generating function $$\sum F_n x^n=\frac{x}{1-x-x^2}.$$

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