# Module MA2325: Complex Analysis I

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Michaelmas term 2012-13
**Contact Hours**- 11 weeks, 3 lectures including tutorials per week
**Lecturer**- Prof. Dmitri Zaitsev
**Learning Outcomes**- Manipulate and calculate with complex numbers, complex functions
(polynomials, rational functions, exponential and trigonometric
functions) and multi-valued functions (argument, logarithm and square
root).
- Identify subsets of the complex plane and their geometric and
topological properties (open, closed, connected, bounded, convex,
star-shaped etc).
- 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
complex function.
Derive the Cauchy-Riemann equations for a complex differentiable
function and identify whether a function is complex differentiable at
a point.
- 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.

- Manipulate and calculate with complex numbers, complex functions
(polynomials, rational functions, exponential and trigonometric
functions) and multi-valued functions (argument, logarithm and square
root).
**Module Content**- Complex numbers.
- Analytic functions.
- Complex integration.
- Power series.
- Residue theorem and applications.

**Recommended Reading:**- Complex variables and applications, J.W. Brown, R.V. Churchill. McGraw-Hill, 2003.
- Complex analysis, L.V. Ahlfors. McGraw-Hill, 1979.
- Complex function theory, D. Sarason. Oxford University Press, 2007.
- Complex analysis, T.W. Gamelin. Springer, 2001.
- Functions of one complex variable, J.B. Conway. Springer-Verlag, 1984.

- Complex numbers.
**Module Prerequisite****Assessment Detail**-
This module will be examined in a 2 hour
**examination**in Trinity term.