Brief Introduction
The course covers the most important topics of complex analysis. We start with the definition of a complex number and progress quickly to the concept of complex derivative and the analytic function of a complex variable. Next, we move to contour integration in the complex plane and discuss vital theorems of complex analysis (such as Cauchy's and Jordan's). We complete the course with the integration of multivalued functions and introducing the concept of Riemann surfaces.
Description
"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.
Knowledge
- The students will learn:
- 1. Major methods and theorems of complex analysis.
- 2. How to Laurent expand functions near singularities.
- 3. How to compute complex and real integrals with the help of Cauchy and residue theorem.
- 4. How to extract regular branches of multivalued functions and compute their values and residues.
- 5. How to perform analytical continuation of multivalued functions with different methods.
- 6. How to build a Riemann surface with bare hands and with the help of Wolfram Mathematica.
- 7. How to compute integrals containing multivalued functions.
- 8. How to compute integrals of analytic functions along contours spanning several Riemann sheets.
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