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Complex Analysis

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.

This course is archived
Estimated 7 weeks
6–12 hours per week
Instructor-led on a course schedule
Optional upgrade available

About this course

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"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.

At a glance

  • Institution: MISISx
  • Subject: Math
  • Level: Intermediate
  • Prerequisites:

    Real analysis, multivariate analysis.

  • Language: English
  • Video Transcript: English

What you'll learn

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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.

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.

Learner testimonials

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"This course is excellent throughout. The videos are done well, with a special format where the text is rendered as the speaker writes."

"I had a great time with the problems. They have a strong geometric flavour, you must find your way along Riemann surfaces' twists and turns, your path or contour all the way stretching and bending around branch points and poles."

"The problem sets are excellent and challenging. The material in this course will be quite useful in theoretical physics."

About the instructors

Who can take this course?

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