Brief Introduction
The course covers the most important topics of complex analysis. We start with the definition of a complex number and finish with the integration of multivalued functions and Riemann surfaces.
Description
The course is practice-oriented. It is supplemented with many problems aimed at assisting the understanding of lecture materials.
Each problem, in turn, is supplemented with a detailed solution.
The topics covered:
1. Complex algebra, complex differentiation, simple conformal mappings.
2. Taylor and Laurent expansion.
3. Residue theory. Integration of contour and real integrals with the help of residues.
4. Multivalued functions and regular branches
5. Analytic continuation and Riemann surfaces.
6. Integrals with multivalued functions.
The course includes two tracks.
The free track allows the learner to access all the materials from the course.
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.
Knowledge
- The students will learn how to:
- 1. Laurent expand functions near singularities.
- 2. Compute complex real integrals with the help of residue theorem.
- 3. Extract regular branches of multivalued functions and compute their values and residues.
- 4. Perform analytical continuation of multivalued functions.
- 5. Build a Riemann surface with bare hands and with the help of Wolfram Mathematica.
- 6. Compute integrals containing multivalued functions.
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