Complex Analysis

10 credits

Syllabus, Bachelor's level, 1MA022

A revised version of the syllabus is available.

Code: 1MA022
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G2F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 12 December 2011
Responsible department: Department of Mathematics

Entry requirements

60 credits of Mathematics including Several Variable Calculus. Alternatively, 40 credits Physics and 40 credits Mathematics including Several Variable Calculus or Geometry and Analysis III.

Learning outcomes

In order to pass the course (grade 3) the student should

be able to give an account of the concepts of analytic function and harmonic function and to explain the role of the Cauchy–Riemann equations;

be able to explain the concept of conformal mapping, describe its relation to analytic functions, and know the mapping properties of the elementary functions;

be able to describe the mapping properties of Möbius transformations and know how to use them for conformal mappings;

know how to define and evaluate complex contour integrals;

know the Cauchy integral theorem, the Cauchy integral formula and some of their consequences;

be able to analyse simple sequences and series of functions with respect to uniform convergence, describe the convergence properties of a power series, and determine the Taylor series or the Laurent series of an analytic function in a given region;

know the basic properties of singularities of analytic functions, how to determine the order of zeros and poles, how to compute residues and how to evaluate integrals using residue techniques;

know how to determine the number of roots in a given area for simple equations;

be able to formulate important results and theorems covered by the course and describe the main features of their proofs;

be able to use the theory, methods and techniques of the course to solve mathematical problems;

be able to present mathematical arguments to others.

Content

Complex numbers, topology in C. Functions of one complex variable, limits, continuity and differentiability. The Cauchy-Riemann equations with consequences. Analytic and harmonic functions. Conformal mappings. Elementary functions from C to C, in particular Möbius transformations and the exponential function, and their mapping properties. Solution of boundary value problems in the plane for the Laplace equation using conformal mappings. Complex integration. Cauchy's integral theorem and integral formula with consequences. The maximum principle for analytic and harmonic functions. Conjugate harmonic functions. Poisson's integral formula. Uniform convergence and analyticity. Power series. Taylor and Laurent series with applications. Zeros and isolated singularities. Residue calculus with applications. The argument principle and Rouché's theorem. Briefly about connections with Fourier series and Fourier integrals.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course