Complex analysis

Complex analysis is the calculus of complex numbers. One might think that the calculus of complex numbers would be quite similar to the calculus of real numbers, but, amazingly, this turns out to be not the case. There are many pathological functions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis.

Cauchy Integral Formula

Let f be holomorphic on a simply connected domain D, and let $\Gamma\subseteq D$ be a simple closed Jordan curve. Then for any $z_0$ in the interior of $\Gamma$, we have \[f^{(n)}(z_0)=\frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}.\] In particular, the value of a holomorphic function inside a region is determined uniquely by its values on the boundary! This is certainly not true of a real function, even a real analytic function.

Liouville's Theorem

Let $f$ be an entire function (i.e. holomorphic on the whole complex plane). If $\lvert f(z)\rvert  \le A$ for all $z$ for some real number $A$, then $f$ is a constant function.

Picard's Little Theorem

This is a powerful generalization of Liouville's Theorem. If $f$ is an entire function so that there exist two complex numbers $a$ and $b$ such that for every complex number, $f(z)\neq a$ and $f(z)\neq b$, then $f$ is a constant function.

See also