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The Fundamental Theorem of Algebra, and the Fundamental Theorem of Analysis

Module by: Lawrence Baggett. E-mail the author

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 ff is a differentiable function of a complex variable on an open set U⊆C,U⊆C, then we will see that ff is actually expandable in a Taylor series around every point in U.U. In particular, a function ffof a complex variable is guaranteed to have infinitely many derivatives on UU if it merely has the first one on U.U. This is in marked contrast with functions of a real variable. See part (3) of (Reference).

The main points of this chapter are:

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

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