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