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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**

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.

### Syllabus

**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

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