Mathematics
Complex Analysis and the Residue Theorem for Evaluating Integrals
Quick fact
The residue theorem states that the integral of a complex function around a closed curve is 2πi times the sum of the residues of the function at its singularities inside the curve. This allows evaluating many definite integrals of real functions that are otherwise intractable, by embedding them in the complex plane and closing a contour.
Why this is interesting
You've likely struggled with integrals like ∫₀^∞ sin(x)/x dx, which don't yield to elementary methods. But there's a technique that turns such 'impossible' integrals into a simple count of certain numbers--residues--using the machinery of complex numbers.