The Connections Project

Complex Analysis

The complex plane is a generalization of the real plane in the most obvious way. The latter is embedded in the former. Many functions in real analysis, such as the exponential function and the sine function, hide profoundly deep properties until they are extended in complex analysis.

The very definition of the complex numbers traditionally comes from abstract algebra: you take the ring of real polynomials, mod out by the ideal generated by x^2+1, and since x^2+1 is irreducible, theorems of algebra guarantee the resulting quotient ring is actually a field-- a field with a square root of -1. Complex analysis generously repays the favor by providing an unexpected proof of the Fundamental Theorem of Algebra (which states that every non-constant polynomial with complex coefficients has a root).

There are actually some startlingly strong connections between these utterly different-looking fields. Most famously, the proof of the Prime Number Theorem is most easily accomplished using complex analytical machinery. The zeta function, about which the famous Riemann Hypothesis is concerned, would offer lots of further insight into prime numbers, if Riemann's hypothesis were proved.