Complex Analysis

"Complex analysis" is a practice-oriented course. Both tracks of the course (audit and verified) are supplemented with carefully chosen problems aimed at assisting the understanding of lecture materials. Each problem, in turn, is supplemented with a detailed solution.

The major concepts of complex analysis have a strong geometric flavor. Therefore, whenever possible we use geometrical interpretation of principal ideas to invoke the spatial intuition of the learner.

The majority of the topics of the course (e.g. Taylor's and Laurent's power series, Cauchy's and residue theorems) are given with immediate examples to sharpen the learner's grasp. The focal point of complex analysis is of course, the art of contour integration in the complex plane.

Building on the concept of analytic function we successively introduce the complex contour integral and main integral theorems. Gradually developing this idea we finish the course with integration along contours spanning several Riemann sheets.

The topics covered:

1. Complex numbers, complex algebra, complex derivative, analytic function, simple conformal mappings.

2. Cauchy theorem. Taylor and Laurent power series.

3. Residue theory. Contour integration. Computation of real integrals with the help of residues. Cauchy principal value integral.

4. Multivalued functions: branch points and branch cuts. The computation of regular branches.

5. Methods of analytic continuation. Analytic continuation with the help of contour deformation. Riemann surfaces of analytic functions.

6. Integrands with multivalued functions.

The course includes two tracks.

The audit track allows the learner to access all lecture materials from the course including many problems.

The "verified certificate" track allows the learner to

1. access additional non-trivial problems from the course.

2. access the detailed solutions to all the problems inside the course at the end of each week.

3. get an official certificate from the university on completion of the course.