Entry requirements

Several Variable Calculus, Limited Version.

Learning outcomes

On completion of the course, 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;
• evaluate complex contour integrals, and apply the Cauchy integral theorem, the Cauchy integral formula and some of their consequences;
• 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;
• give an account of the concept singularity, determine the order of zeros and poles, compute residues and 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;
• 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. 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 and assignments given during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

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