2024/2025
Специальные функции
ID 905384
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Хорошкин Сергей Михайлович
Язык:
английский
Кредиты:
6
Course Syllabus
Abstract
The course offers a practice-oriented introduction to the special functions of mathematical physics, primarily those of the hypergeometric type. Its aim is to develop proficiency in the standard analytical methods used to study special functions through classical examples: (1) representing functions in the form of integrals and series, (2) determining and applying asymptotic behavior, (3) evaluating special values and degenerations, (4) studying differential and difference properties, and so on. The functions considered (the gamma function, zeta functions, Gauss hypergeometric function, etc.) are somehow at the next level of complexity after elementary functions. These special functions and the methods used to study them are crucially important for mathematics, theoretical and mathematical physics, and physical chemistry.
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