Elementary functions in complex arithmetic

Abstract

Since the invention of complex arithmetic about two centuries ago, mathematicians have learned how to extend functions from the real line to the complex plane. The field of complex analysis is now well understood, and mathematics courses in that area are commonly offered in the first years of undergraduate college eduction.

One of the most profound jokes of nature is the square root of minus one that physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926. . . . the Schrödinger equation describes correctly everything we know about the behavior of atoms. it is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.

— Freeman Dyson Birds and Frogs (2008 AMS Einstein Lecture).