The fundamental theorem of algebra, and the fundamental theorem of

In this chapter we will discover the incredible difference between the analysis of functionsof 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 factthey are extremely different in character. Indeed, if $f$ is a differentiable function of a complex variable on an open set $U\subseteq C,$ then we will see that $f$ is actually expandable in a Taylor series around every point in $U.$ In particular, a function $f$ of a complex variable is guaranteed to have infinitely many derivatives on $U$ if it merely has the first one on $U.$ This is in marked contrast with functions of a real variable. See part (3) of [link] .

The main points of this chapter are:

1. The Cauchy-Riemann Equations ( [link] ),
2. Cauchy's Theorem ( [link] ),
3. Cauchy Integral Formula ( [link] ),
4. A complex-valued function that is differentiable on an open set is expandable in a Taylor series around each point of the set ( [link] ),
5. The Identity Theorem ( [link] ),
6. The Fundamental Theorem of Algebra ( [link] ),
7. Liouville's Theorem ( [link] ),
8. The Maximum Modulus Principle (corollary to [link] ),
9. The Open Mapping Theorem ( [link] ),
10. The uniform limit of analytic functions is analytic ( [link] ), and
11. The Residue Theorem ( [link] ).