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 Theorem
Let f be holomorphic on a simply connected domain D, and let 
be a simple closed Jordan curve. Then for any 
in the interior of 
, we have
![\[f^{(n)}(z_0)=\frac{n!}{2\pi i} \int_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}.\]](http://latex.artofproblemsolving.com/9/f/9/9f9046b07355887d24ebed4c5afa1358588c4a6b.png)
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 
be an entire function (i.e. holomorphic on the whole complex plane). If 
for all 
for some real number 
, then is a constant function.
Picard's Little Theorem
This is a powerful generalization of Liouville's Theorem. If is an entire function so that there exist two complex numbers 
and 
such that for every complex number, 
and 
, then is a constant function.
