Complex Analysis is the theory of functions of a single complex variable.  (That is: Functions that map from complex numbers to complex numbers.)  This is advanced undergraduate mathematics with applications to even more advanced mathematics, physics, and engineering.  Some of the topics covered:
 
Fred's scheduling issues
Cauchy-Riemann equations; Cauchy-Goursat theorem; Cauchy’s formula
Derivatives and Integration in the set of complex numbers
Casorati-Weierstrass theorem; Picard’s theorems; Liouville theorem
Why the Residue theorem is called a residue
Red Riding Hood