The

calculus (

i.e. analysis) of

complex
numbers. It is quite advanced compared to ordinary

calculus which
uses

real numbers, mainly because complex numbers lie on a
two-

dimensional

plane, instead of a line of real numbers.

For instance, with real variables the integral of a function from
z_{1} to z_{2} is well defined along a unique
path. But on the complex plane, there are infinitely many paths from
one number to another, and the integral may depend on the choice of
the path. Similar complications exist with complex derivatives. There's a good reason why these numbers are called *complex* ;-).

On the plus side, complex analysis has powerful results which can
simplify some calculations enormously. For example:

Among other uses, complex analysis provides powerful tools for
physics in a two-dimensional system, examples being fluid dynamics
and electromagnetism. For a higher number of dimensions, there are
similar techniques in Geometric algebra.