On successful completion of this module, students will be able to:
Manipulate and calculate with complex numbers, complex functions
(polynomials, rational functions, exponential and trigonometric
functions) and multi-valued functions (argument, logarithm and square
Identify subsets of the complex plane and their geometric and
topological properties (open, closed, connected, bounded, convex,
Determine if a sequence of complex numbers is convergent, compute
the limit of a given sequence and apply the Cauchy criterion.
Define the limit of a complex function at a point and apply
properties of limits. Compute the limit of a complex function at a
point and determine whether a given complex function is continuous.
Define the derivative of a complex function, state and prove
properties of the derivative and compute the derivative of a given
Derive the Cauchy-Riemann equations for a complex differentiable
function and identify whether a function is complex differentiable at
Determine if an infinite series of complex numbers is convergent.
Describe the convergence properties of a complex power series, derive
formulae for and compute the radius of convergence.
Identify and construct examples of paths satisfying prescribed
properties. Evaluate complex path integrals and state and prove
properties of such integrals. Define the index function for a path,
describe its properties and evaluate winding numbers.
State and prove versions of Cauchy's theorem and its consequences
including Cauchy's integral formula, the power series representation
for analytic functions, Liouville's theorem and the Fundamental
Theorem of Algebra.
Find Taylor and Laurent series for a complex function, compute
residues and apply the residue theorem to evaluate integrals.