The theory of functions of a complex variable

Complex numbers, complex functions, analytic functions, conformal mapping, operational calculus, the use of complex analysis methods in theoretical and mathematical physics, applied aspects of complex analysis in physics and other natural sciences.

Read at 3-rd semester.
2 hours of lectures and 2 hours of seminars per week

• Kravtsov Andrey Vladimirovich
• Nedelko Ilya Vitalevich
• Popov Viktor Yrevich

1. Complex numbers, functions of a complex variable.
2. Continuity and differentiability of functions of a complex variable. . The notion of analytic functions of a complex variable.
3. The integral of the function of a complex variable, Cauchy-type integral. Properties of integrals.
4. Rows of analytic functions. Power series.
5. The notion of analytic continuation. Elementary functions of a complex variable as the analytic continuation of functions of a real variable.
6. Laurent series. The singular points of functions.
7. Deductions. The fundamental theorem of the theory of residues. Calculation of improper integrals of a real variable by means of residues. The fundamental theorem of Algebra.
8. Conformal mapping. The main functions used in conformal mappings. Some applications of conformal mappings.
9. The basic concepts of the operational calculus.
10. The method of steepest descent.

References:

1. A.G.Sveshnikov, Tikhonov. The theory of functions of a complex variable. Moscow: Publishing House "Nauka", 1999.
2. L.I.Volkovysky, G.L.Lunts, I.G.Aramanovich. Problems in the theory of functions of a complex variable. M. FIZMATLIT, 2004.
3. A.V.Kravtsov, A.R.Maykov. Guide to the course in the theory of complex functions. Moscow: Physics Department of Moscow State University, 2007.

Further reading:

1. M.L. Krasnov, A.I. Kiselev, G.I. Makarenko. Operational calculus. The theory of stability. The tasks and examples with detailed solutions Moscow: URSS in 2003.
2. M.L. Krasnov, A.I. Kiselev, G.I. Makarenko, E.V. Shikin, V.I. Zalyapin, S.K. Sobolev. All higher mathematics v.4 Moscow: URSS, 2001.
3. M.A. Lavrentiev B.V. Shabat. Methods of the theory of functions of a complex variable. Moscow: Publishing House "Nauka", 1987.
4. Yu.V. Sidorov, M.V. Fedoryuk, M.I. Shabunin. Lectures on the theory of functions of a complex variable. Moscow: Publishing House "Nauka", 1976.

• Manual to the tests and examinations of complex analysis in 2015
• The course program (2009-2010) //105Кб 23.09.2009//
• Questions of the overall standings (2012-2013)
• Questions to the first part of the exam (2012-2013)
• Questions to the second part of the exam (2012-2013) (questions on 4 and 5)