Functions of Complex Variables
- Department of Applied Mathematics
Complex functions of complex variables, Cauchy theorem, residues, holomorphic and meromorphic functions, Taylor and Laurent series, analytic functions.
- Syllabus of lectures:
1.Complex numbers and their representation in Cartesian and polar form. Principal value of argument arg z and multivalued function Arg z. Gauss' complex plane C*.
2.Progressions and series of complex numbers. Limits and convergence of progressions and sum of series.
3.Complex functions of complex variable real and imaginary parts. Limits and continuity of complex functions.
4.Elementary complex functions, definition and properties.
5.Multi-valued complex functions, definition and properties.
6.Differentiation of complex functions of complex variable. Cauchy - Riemann equations. Holomorphic function.
8.Classification of singularities: Zeros and poles, multiplicity.
9.Progressions and series of complex functions, Weierstrass' criterion of convergence, differentiation and integration of power expansions of holomorphic functions.
10.Taylor series, summation.
11.Laurent series, residua.
12.Path integral. Cauchy's integral formula.
13.Residue theorem and its application.
14.Meromorphic functions, analytic function.
- Syllabus of tutorials:
- Study Objective:
- Study materials:
Conway J. B.: Functions of one complex variable, 1. -- 2. ed., New York, Springer, 1978
Conway J. B.: Functions of one complex variable, 2. - 2. ed., New York, Springer, 1996
Bugrov J. S., Nikoĺskij S. M.: Differential equations, multiple integrals, series, theory of functions of a complex variable, 1. ed., Moscow, Mir, 1983
Sidorov J. V., Fedorjuk M. V., Šabunin M. I.: Lectures on the theory of functions of a complex variable, 1. ed., Moscow, Mir, 1985
- Further information:
- No time-table has been prepared for this course
- The course is a part of the following study plans: