MATH2021
Complex Analysis
10 credits

Taught Semester 2 View Timetable

Year running 2006/07

Pre-requisites
MATH2011, or equivalent. Exclusion: Not with MATH2090

This module is approved as an Elective

Module summary
Prerequisite: MATH2011 or equivalent
Excluded combination: may not be taken with MATH2090
Informal description: : Complex analysis was the great triumph of nineteenth century mathematics. The work of the French mathematician Cauchy laid the foundations for many deep results and applications to other branches of mathematics. The latter part of this course is an exposition of Cauchy's beautiful and surprising theorems about analytic functions. One such result enables us to use integration in the complex plane to calculate definite integrals which apparently do not involve the complex numbers at all!
See the schools website or contact: a.slomson@leeds.ac.uk for more information.

Objectives
General introduction to complex analysis, which is central to Pure Mathematics, with applications relevant in Applied Mathematics. On completion of this module, students should be able to: (a) use the Cauchy-Riemann equations to decide where a given function is analytic; (b) compute the harmonic conjugates of typical harmonic functions; (c) determine the radius of convergence of complex power series; (d) compute standard contour integrals using the fundamental theorem of the calculus, Cauchy?s theorem or Cauchy?s integral formula; (e) give a complete proof of the fundamental theorem of algebra; (f) classify the singularities of analytic functions and to compute, in the case of a pole, its order and residue; (g) evaluate typical definite integrals by using the calculus of residues.

Syllabus
This module treats the elementary theory of complex-valued functions. It turns out that a very elegant theory can be built up using the geometry of plane curves. The module has many useful applications, including the fundamental theorem of algebra (that every complex polynomial has a root), as well as conformal mappings, harmonic functions and contour integration, which are the basic techniques in applied mathematics. 1. Complex sequences and series. Convergence of sequences and series. Complex exponential and logarithm. 2. Analytic functions. Open sets in C. Continuous functions on open sets. Analytic functions; definition, algebra of analytic functions. Entire functions. Cauchy-Riemann equations. Harmonic functions. Power series and discs of convergence. 3. Contour integration. Definitions of contours and closed contours. Integrals of continuous functions along a contour. Estimates for integrals. Fundamental theorem of the calculus for analytic functions. 4. Cauchy's theorem and integral formula. Winding number. Cauchy's theorem Cauchy's integral formula. Liouville's theorem for entire functions. Maximum modulus theorem. Fundamental theorem of algebra. Schwarz's lemma. 5. Taylor's theorem. Formula for coefficients in complex Taylor series. Analytic functions are infinitely differentiable. 6. Calculus of residues. Definitions of pole of order m, simple pole, removable singularity, essential singularity, residue. Cauchy's residue theorem. Application to calculation of definite integrals; Jordan's lemma, summation of series.

Teaching methods
Lectures (22 hours) and examples classes (11 hours).

Methods of assessment
2 hour written examination at end of semester (100%).

Reading list
The reading list is available from the Library website

Last updated: 30/03/2007

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