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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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    $\begingroup$ 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. $\endgroup$
    – Sam
    Commented Jul 13, 2012 at 22:05

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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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The first and simplest proofs of the Prime Number Theorem rely on showing that the (analytic continuation of the) Riemann zeta function has no zeros on the line, real part equals 1.

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In operator theory, complex contour integrals called Dunford-Taylor integrals provide a functional calculus for operators on function spaces. This serves as the foundation for many purposes in the theory of matrix analysis, PDE, numerical analysis, and control theory, just for a start.

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