Introduction to elliptic functions

General data

Course ID:	1000-1M10EF
Erasmus code / ISCED:	11.134 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Introduction to elliptic functions
Name in Polish:	Introduction to elliptic functions
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Prerequisites (description):	ODEs, analysis
Short description:	Elliptic functions (doubly periodic functions) have many applications in modern mathematis (ODEs, number theory, algebraic geometry, topology and others). In the course we start with elliptic integrals, then continue with an analytic description of elliptic functions via first order nonlinear ordinary differential equation. Further topics include elliptic curves and applications to special functions. We shall deal with concreto problems and examples.
Full description:	Starting from the works of Gauss, Cauchy, Abel, Jacobi, Eisenstein, Riemann, Weierstrass, Klein and Poincare, the theory of meromorphic functions of a complex variable has developed significantly and has direct links with analysis, differential equations, algebra, number theory, potential theory, geometry and topology. It makes an interesting and important topic for study. Since the theory is very rich, we shall mainly concentrate on the analytic viewpoint in the course. Connections with other areas will be discussed as well. We shall start with comprehensive introduction to elliptic functions, which are doubly periodic functions. They arose from attempts to evaluate certain integrals associated with the formula for the circumference of an ellipse. They can be regarded as meromorphic functions on the torus. Moreover, elliptic functions are the rational functions of the Weierstrass function and its derivative, these two functions being related by a first order nonlinear ordinary differential equation. Other topics that could be covered include automorphic functions, applications to number theory. If time permits, other special functions like hypergeometric functions and their confluences, Lame functions etc will be discussed. Prerequisite courses: ordinary differential equations, complex analysis.
Bibliography:	T. Ekedahl, One semester of elliptic curves (available online in BUW) N. Akhiezer, Elements of the theory of elliptic functions Additonal literature: J. V. Armitage and F. Eberlein, Elliptic functions G. Jones, D. Singermann Complex functions: an algebraic and geometric viewpoint. Batemann, Erdelyi, Higher transcendental functions T. Apostol, Modular functions and Dirichlet series in number theory. and other books available online in BUW
Learning outcomes:	The basic knowledge of the theory of elliptic functions, elliptic integrals and elliptic curves.
Assessment methods and assessment criteria:	Written exam or a presentation of project

This course is not currently offered.

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