Web page of Dmytro Savchuk
Complex Analysis is a fundamental field of mathematics that, in a quite literal sense, `completes' what we can do with real numbers. Roughly, the complex numbers are formed from the real numbers by adding an extra number "the square root of -1" to be able to state answers for some previously unanswerable problems. Many of the tools from calculus on real numbers can be generalized to apply to the complex numbers, but there are also many differences between the reals and the complexes. This is especially apparent in many amazingly powerful tools for complex numbers which have no real analogue.
The goal of this course is to explore the complex numbers and many results which work with them. We will generalize calculus to deal with differentiating and integrating functions defined over the complex numbers. We will then concentrate on a few big hammers in complex analysis: Cauchy's Theorem, harmonic functions, Taylor series, and the Residue Theorem.
December 31st, 1969
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