Module overview
Complex Analysis is the theory of functions in a complex variable. While the initial theory is very similar to Analysis (i.e, the theory of functions in one real variable as seen in the second year), the main theorems provide a surprisingly elegant, foundational and important insight into Analysis and have far-reaching and enlightening applications to functions in real variables.
After introducing differentiability of complex functions and contour integrals, the highlights of this module are presented: Cauchy’s Theorem, Cauchy’s Integral Formula and the Residue Theorem. Finally, the theory is applied to prove the Fundamental Theorem of Algebra, to evaluate real integrals, to provide partial-fraction and infinite-product expansions of classical functions and to introduce and study the Gamma function (a fundamental function that for example affords an asymptotic formula for factorials and a computation of the integral of the Gaussian bell curve).
Linked modules
Pre-requisites: MATH1048 AND MATH2039