Special Functions
Lectures: Wednesday 13-15 and Friday
12-14 in auditorium 10
Textbook: "Special Functions" by G. Andrews, R. Askey
and R. Roy (Cambridge University Press)
Exam: During the semester there
will be four problem sets to solve. To get credit for the course all
four sets have to be approved. The last set is a take home exam which
will be graded upon request.
The
Gamma and Beta Functions
Textbook: Section
1.1+1.2 (÷Thm 1.2.7)+1.3 (÷Thm 1.3.4)+1.5+1.9
Copies: Infinite
products (4), Uniqueness theorems (2), Bernoulli polynomials and
Stirling's formula (15)
September
3: Talked about Section 1.1 including some basic results about
infinite products corresponding to Appendix A. Copies of four pages
about infinite products were handed out.
September 5: Talked
about Section 1.2 (except Theorem 1.2.7) and the first part of Section
1.5. Section 1.3 was skipped since we later on consider the Riemann zeta
function in more detail. Copies of a paper about the history of the
gamma function were handed out.
September 10: Finished
the proof of Theorem 1.5.2 and talked about the Bohr-Mollerup theorem
from Section 1.9 and Wielandt's theorem. Copies of two pages about
uniqueness theorems were handed out. Stirling's formula from Section 1.4
was proved in the special case x>0. Section 1.6 - 1.8 was skipped.
September 12: Talked
about the Bernoulli polynomials, Stirling's series and Stirling's
formula in the cut plane with the negative half-line removed. Copies of
fifteen pages of handwritten notes were handed out.