Summary: An introduction to the fundamental theorem of algebra, with links to important theorems on the topic.

In this chapter we will discover the
incredible difference between the analysis of functions
of a single complex variable as opposed to functions of a single
real variable.
Up to this point, in some sense, we have treated them as being
quite similar subjects, whereas in fact
they are extremely different in character.
Indeed, if

The main points of this chapter are:

**The Cauchy-Riemann Equations**((Reference)),**Cauchy's Theorem**((Reference)),**Cauchy Integral Formula**((Reference)),**A complex-valued function that is differentiable on an open set is expandable in a Taylor series around each point of the set**((Reference)),**The Identity Theorem**((Reference)),**The Fundamental Theorem of Algebra**((Reference)),**Liouville's Theorem**((Reference)),**The Maximum Modulus Principle**(corollary to (Reference)),**The Open Mapping Theorem**((Reference)),**The uniform limit of analytic functions is analytic**((Reference)), and**The Residue Theorem**((Reference)).