Entry requirements

Several Variable Calculus, limited version

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;
• know how to evaluate complex contour integrals, and be familiar with the Cauchy integral theorem, the Cauchy integral formula and some of their consequences;
• be able to describe the convergence properties of a power series and to 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;
• 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. Complex integration. Cauchy's integral theorem and integral formula with consequences. Power series. Uniform convergence and analyticity. Laurent series with applications. Zeros and isolated singularities. Residue calculus with applications. The argument principle and Rouché's theorem.

Instruction

Lectures and problem solving sessions.

Assessment

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

Other directives

The contents of this course is part of Complex Analysis (1MA022), 10 credit points. Both courses cannot be credited for in diploma.