Complex analysis

Курс "Комплексный анализ" предлагает изучение алгебры комплексных чисел, дифференцирования и интегрирования функций комплексных переменных, а также условий Коши-Римана и конформных отображений. Уделяется внимание рядам Тейлора и Лорана, теории вычетов, аналитическому продолжению и поверхностям Римана. Студенты научатся работать с многозначными функциями и решать сложные интегралы

Lavrentev M.A., Shabat B.V. Methods of the theory of function of complex variable. Nauka. Moscow, 1987

1. Week 1: Algebra of complex numbers. Integration and differentiation of functions of complex variables.

2. Geometric, trigonometric and exponential representation of a complex number. Differentiation of functions of complex variables. Cauchy-Riemann conditions. Introduction to conformal mappings. Integration.

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

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

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

6. Integration with residues I. Residue at infinity. Integration with residues II. Jordan's lemma. Integration with Jordan's lemma. Integration in principal value.

7. Week 4: Multivalued functions and regular branches.

8. Extraction of the regular branch of the power type function [theory]. Extraction of the regular branch of the power type function [example]. Extraction of the regular branch of the log function. Practice with regular branches.

9. Week 5: Analytical continuation and Riemann surfaces.

10. Analytical continuation. Simple example. Formal definition and uniqueness of analytic continuation. Practice with analytic continuation: contour deformation. Riemann surfaces [theory]. Riemann surfaces [example].

11. Week 6: Integrals containing multivalued functions.

12. 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.

13. Week 7: Final Exam

В результате освоения курса у обучающихся формируются следующие компетенции:

• good command of handling contour integrals and real-valued integrals

• good knowledge in the practical applications of essential theorems of complex analysis.

• firm knowledge of the theory of multivalued function

• the introductory knowledge of the theory of Riemann surfaces

Integration in the complex plane

Integration on the Riemann surface (introduction)

Work with multivalued functions