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Complex Analysis

National University of Science and Technology MISiS via edX

Overview

"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.

Syllabus

Lecture 1

  • Representations of complex numbers.
  • Complex derivative. Cauchy-Riemann conditions
  • Simple conformal mappings.

Lecture 2

  • Cauchy integral theorem.
  • Taylor and Laurent series in the complex plane
  • Types of singularities

Lecture 3

  • Integration with residues
  • Integration with Jordan's lemma
  • Integration in principal value

Lecture 4

  • Extraction of the regular branch of the power and log-type function.

Lecture 5

  • Analytical continuation. Interesting examples.
  • Riemann surfaces.

Lecture 6

  • Integrals with power and log-type integrand.
  • Integrals along contours lying on different Riemann sheets.

Taught by

Yaroslav Rodionov and Konstantin Tikhonov

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Reviews

5.0 rating, based on 4 reviews

Start your review of Complex Analysis

  • This excellent course is sort of a second course in the complex analysis in that it emphasizes aspects like multivalued functions, analytic continuation, and Riemann surfaces that are often neglected in first courses, and often receive only a page or two in textbooks. The problem is that the textbooks cover trivial examples (square root) and don't go into much detail, whereas this course really goes into the details.

    The problem sets are excellent and challenging. The material in this course will be quite useful in theoretical physics.
  • Niekrakhorst
    Very much recommended ! The first half of the course is a useful review of standard material; at times maybe a bit terse for those at first exposure. The second half was fascinating. I had a great time with the problems. They have a strong geometric...
  • Profile image for James Freericks
    James Freericks
    This course is a Russian style complex analysis course. While it starts from the basics, assuming only multivariable calculus, you must be quite skilled in math to make it through if you have never seen complex analysis before. What is unique about this...
  • Anonymous
    thanks to Yaroslav for the excellent course, but I would like a little more theory on special functions and their application =)

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