There are two types of numbers in mathematics, real and imaginary. Complex numbers have both a real and imaginary part, with complex analysis dealing with such numbers and their functions.

The question that naturally arises among students is: how do complex numbers help us in real life? It turns out that there are many important examples of this, a few of which are given with a simple discussion at the Question Corner from the University of Toronto Mathematics Network.

The owner of had personal experience with the importance of complex numbers during his own research. See the blog article Reflections of a Ph.D..

In a typical course in complex analysis, the following topics will be covered:

  • Complex numbers
  • Complex functions
  • Elementary functions
  • Integration
  • Cauchy's Theorem
  • Cauchy's Integral Formula
  • Liouville's Theorem
  • Maximum moduli
  • Harmonic functions
  • Series
  • Taylor and Laurent Series
  • Poles, residues
  • Argument principle
  • Rouche's Theorem

There are many good books on complex analysis. In addition, the International Journal of Open Problems in Complex Analysis should not be overlooked.

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