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;
• explain the concept of conformal mapping, describe its relation to analytic functions, and know the mapping properties of the elementary functions;
• describe the mapping properties of Möbius transformations and know how to use them for conformal mappings;
• define and evaluate complex contour integrals;
• give an account of and use the Cauchy integral theorem, the Cauchy integral formula and some of their consequences;
• 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;
• give an account of the basic properties of singularities of analytic functions and be able to determine the order of zeros and poles, to compute residues and to evaluate integrals using residue techniques;
• determine the number of roots in a given area for simple equations;
• formulate important results and theorems covered by the course and describe the main features of their proofs;
• use the theory, methods and techniques of the course to solve mathematical problems;
• 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 according to instructions delivered at course start.

Other regulations

The course can not be included in higher education qualification together with the course Complex Analysis (1MA021), 5 credits.