In this module, you will extend your understanding of differential and integral calculus from functions of a real variable to functions of a complex variable. The differences between the two are often unexpected and very surprising. You will develop the theory with applications such as evaluating path integrals via residue calculus, the fundamental theorem of algebra, and conformal mappings.

Topics include:

Algebraic properties of complex numbers

Definition of the derivative, Cauchy-Riemann equations

Power series, radius of convergence

Logarithmic, exponential, and trigonometrical functions, branch points

Line integrals, the Cauchy integral theorem, and integral formula

The Cauchy formula for derivatives, Taylor series

Liouville's theorem, fundamental theorem of algebra

Laurent's theorem, poles and the residue theorem, zeros of analytic functions