About this courseSkip 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 learnSkip 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.
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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Who can take this course?
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