• Length:
    7 Weeks
  • Effort:
    6–12 hours per week
  • Price:

    Add a Verified Certificate for $49 USD

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  • Course Type:
    Instructor-led on a course schedule


Real analysis, multivariate analysis.

About this course

Skip About this course

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.

What you'll learn

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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.

Lecture 1: Algebra of complex numbers.

  • Integration and differentiation of functions of complex variables
  • Geometric interpretation of a complex number
  • Trigonometric representation of a complex number
  • Exponential representation of a complex number
  • Practice with an exponential representation of a complex number
  • Differentiation of functions of complex variables. Cauchy-Riemann conditions
  • Practice with Cauchy-Riemann conditions
  • Introduction to conformal mappings. Integration

Lecture 2: Cauchy theorem. Types of singularities. Laurent and Taylor series.

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

Lecture 3: Residue theory with applications to computation of complex integrals.

  • Integration with residues I
  • Residue at infinity
  • Jordan's lemma
  • Integration with Jordan's lemma
  • Integration in principal value

Lecture 4: Multivalued functions and regular branches.

  • Extraction of the regular branch of the power type function
  • Extraction of the regular branch of the log function
  • Practice with regular branches

Lecture 5: Analytical continuation and Riemann surfaces.

  • More on analytical continuation. Simple example
  • Formal definition and uniqueness of analytic continuation
  • Practice with analytic continuation: contour deformation
  • Riemann surfaces [theory]
  • Riemann surfaces [example]

Lecture 6: Integrals containing multivalued functions.

  • Integrals with power-type integrand and two branch points.
  • Integrals with log-type function
  • The second type of integrals with the log-function
  • Integrals with asymmetric integrand and log function

Meet your instructors

Yaroslav Rodionov
Associate Professor
National University of Science and Technology MISIS
Konstantin Tikhonov
Researcher, Theoretical Physics
Landau Institute

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Who can take this course?

Unfortunately, learners from one or more of the following countries or regions will not be able to register for this course: Iran, Cuba and the Crimea region of Ukraine. While edX has sought licenses from the U.S. Office of Foreign Assets Control (OFAC) to offer our courses to learners in these countries and regions, the licenses we have received are not broad enough to allow us to offer this course in all locations. edX truly regrets that U.S. sanctions prevent us from offering all of our courses to everyone, no matter where they live.