Special functions

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Elementary Lie theory of matrix groups and algebras. Fundamental concepts of the theory of representations of compact Lie groups. Representations of the group SU(2).
Special functions as representative functions on compact matrix groups. Characters of representations. Orthogonality relations. Peter-Weyl theorem.
Spherical harmonics as the representative functions of the group SU(2). Legendre polynomials and their properties.
Laplace operator in various coordinate systems. Laplace equation and spherical harmonics. Bessel functions.
Complex differential equations. Riemann and hypergeometric equations. Hypergeometric function. Relation between the hypergeometric function and spherical harmonics.
Linear differential operators. Generalized Fourier series and weak solutions. Differential operators of the Sturm-Liouville type and the associated eigenproblems.

Readings

J. Dieudonné: Special Functions and Linear Representations of Lie Groups, AMS, Providence, 1979.
T. Bröcker, T. T. Dieck: Representations of Compact Lie Groups, Springer, New York, 1985.
E. Zakrajšek: Analiza III, DMFA-založništvo, Ljubljana, 2002.
F. Križanič: Navadne diferencialne enačbe in variacijski račun, DZS, Ljubljana, 1974.
S. Helgason: Invariant Differential Operators and Eigenvalue Representations, v Representation Theory of Lie Groups, Cambridge Univ. Press, Cambridge, 1980.

Objectives and competences

In the course some important classes of special functions are introduced. Some important applications of these functions in mathematics and physics are described. Special functions are considered from three different viewpoints: from the viewpoint of the representation theory of Lie groups, through the theory of differential equations and by means of the theory of differential operators and their eigenproblems. Fundamental concepts of the theory of complex differential equations with the emphasis on the hypergeometric equation are presented.

Intended learning outcomes

Knowledge and understanding: Familiarity with the most important classes of special functions. Understanding some important applications of these functions. Special functions provide a setting where elements of various mathematical fields merge into a unique theory.
The fundamental importance of the notion of symmetry in the theory of differential equations is discussed.
Application: Solving of some advanced mathematical and physical problems whose solutions cannot be expressed in terms of the elementary functions.
Reflection: Mastering the theory through its applications. Understanding various connections among different mathematical theories.
Transferable skills: Ability to use a vast variety of special functions and of the related differential equations in solving mathematical and non-mathematical problems. Students extend their horizon beyond the relatively limited realm of the elementary functions.

Learning and teaching methods

Lectures, classes, seminar projects, homework, consultations

Assessment

Seminar project
Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Miran Černe:
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. Generalized Ahlfors functions. Transactions of the American Mathematical Society, ISSN 0002-9947, 2007, vol. 359, no. 2, str. 671-686. [COBISS-SI-ID 14227801]
ČERNE, Miran, FLORES, Manuel. Quasilinear [overline{partial}]-equation on bordered Riemann surfaces. Mathematische Annalen, ISSN 0025-5831, 2006, vol. 335, no. 2, str. 379-403. [COBISS-SI-ID 13970777]
Janez Mrčun:
MRČUN, Janez. Functoriality of the bimodule associated to a Hilsum-Skandalis map. K-theory, ISSN 0920-3036, 1999, let. 18, št. 3, str. 235-253. [COBISS-SI-ID 9163353]
MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0. [COBISS-SI-ID 12683097]
MOERDIJK, Ieke, MRČUN, Janez. Lie groupoids, sheaves and cohomology. V: EuroSchool PQR2003 on Poisson geometry, deformation quantisation and group representations, Université Libre de Bruxelles, June 13-17, 2003. GUTT, Simone (ur.), RAWNSLEY, John Howard (ur.), STERNHEIMER, Daniel (ur.). Poisson geometry, deformation quantisation and group representations, (London Mathematical Society lecture note series, ISSN 0076-0552, 323). Cambridge [etc.]: Cambridge University Press, cop. 2005, str. 147-272. [COBISS-SI-ID 13657689]
Pavle Saksida:
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]
SAKSIDA, Pavle. Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity, ISSN 0951-7715, 2001, vol. 14, no. 5, str. 977-994. [COBISS-SI-ID 10942809]
SAKSIDA, Pavle. On zero-curvature condition and Fourier analysis. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2011, vol. 44, no. 8, 085203 (19 str.). [COBISS-SI-ID 15909465]