2024/2025
Special Functions
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Sergey Mikhailovich Khoroshkin
Language:
English
ECTS credits:
6
Course Syllabus
Abstract
The course suggests an accessible introduction to the theory of special functions of hypergeometric type. In particular, it concernes with Gauss hypergeometric function and functions given by transormations of degenerate hypergeometric function (Bessel dunction, Airy function, etc.) as well as their further generalizations: basic (q) hypergeometric series, elliptic hypergeometric functions and integrals. Nest to elementary functions, they are in the knowledge base of an educated mathematician, physicist, and chemist. The study of the properties of special functions reveals the elegance of methods that combine the means of real and complex analysis, differential and difference equations.
Course Contents
- Euler’s Gamma function and related integrals. Riemann zeta function
- Classical hypergeometric function: integral representations, hypergeometric identities, adjacency relations, orthogonal Jacobi polynomials, Riemann hypergeometric equation
- Degenerate hypergeometric equation. Asymptotic properties of solutions. Whittaker, Legendre, Airy, Bessel functions.
- Multiple Barnes gamma functions. Double sine and «quantum dilogarithm».
- Special functions in the theory of representations of Lie groups
- Hypergeometric integrals. Selberg and Gustafson integrals. Rhines integral relations
Interim Assessment
- 2024/2025 4th moduleSeminar work 4, first test 2, second test 2, exam 5. If the total score exceeds 10, the result is reduced to 10.
Bibliography
Recommended Core Bibliography
- A course of modern analysis : an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Whittaker, E. T., 2006
Recommended Additional Bibliography
- Специальные функции, Аски, Р., 2013