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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
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 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.
Module Content
Aims to introduce complex variable theory and reach the residue theorem, applications of that to integral evaluation. See http://www.maths.tcd.ie/~zaitsev/2325-2012/2325.html
  • 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.
Module Prerequisite
 
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term.