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