Special Functions

Blok 2A November 13, 2006-January 26, 2007

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The Study Board in the Mathematical Sciences

If you want to find information about the mathematicians you hear about in the course, then look at MacTutor History of Mathematics Archive

The course is taught by Christian Berg . < berg@math.ku.dk> Information about the course will be given from this page. Roughly speaking, the first two weeks will cover the Gamma function, the next two weeks the Riemann zeta function. Then we will discuss the hypergeometric function for 1-2 weeks and use the rest of the time with elliptic functions.

The officiel desciption of the course Concerning evaluation we repeat from the official homepage: Assessment: The grading will be based on three problem sets of which the two best will count 25%+25%. The project will count 50%. Reevaluation: Already passed parts of the ordinary evaluation are not reevaluated, but are part of the assessment with the weight they had in the ordinary evaluation. Other parts are replaced by a 30 minutes oral exam.

The grades are available on January 26, 2007 grades

Schedule: Lectures: Monday 15.15-17.00 Thursday 10.15-12.00 and 13.15-14.00 in Auditorium 7 First time Monday November 13.

Teaching material: I will hand out written material during the course. There is a number of books on special functions which can be consulted as supplementary reading during the course.
Books on Special functions in general:

George Andrews, Richard Askey, Ranjan Roy: Special functions. Cambridge University Press 1999 (paperback edition 2000).
Earl D. Rainville, Special functions, 1960.
I.S. Gradshteyn, I.M. Ryszik, Table of Integrals, series and products, 1965
A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions Vol. I-III, 1953-1955 (The Bateman Project)

The two first books are regular treatments with proofs. The last two are mainly a collection of all the formulas known at the time of writing with indication of where to find the proofs.
Books only on the Gamma function:

E. Artin, The Gamma function, 1964
F. Lösch, F. Schoblik, Die Fakultät (Gammafunktion) und verwandte Funktionen mit besonderer Berücksichtigung ihrer Anwendungen. 1951
N. Nielsen, Handbuch der Theorie der Gamma funktion, 1906.

Problems to be solved.
First problem set: Deadline November 30, 2006: problem1.pdf
Second problem set: Deadline December 14, 2006: problem2.pdf
Project catalogue containing 5 possible projects and literature. It is not available at the web. Deadline January 15, 2007.
Third problem set: Deadline January 22, 2007: problem3.pdf

Week 46: We will study the Gamma function based on the treatment of Artin.
Material handed out on Nov. 13:
Philip J. Davis: Leonohard Euler's integral: A historical profile of the gamma function. Amer. Math. Monthly 66 (1959), 849-869.
Some lecture notes: Handwritten notes pages 1-4 on convex functions. Pages 10-17 of Artin's book. You are supposed to read yourself the article by Davis. I will start on Thursday after Theorem 2.1.
Material handed out on Nov. 16: Pages 18-29 of Artin's book.
I lectured today on pages 17-27. Note that the function Psi(x)=Gamma'(x)/Gamma(x) is called the Digamma function in many books.

Week 47: I will discuss the pages 28-29 of Artin's book, and then I will start on some handwritten notes to be handed out: §6: Bernoulli numbers, Bernoulli polynomials and Stirling's series. If some of you have problems with lack of knowledge of complex function theory, then look at either

Stewart and Tall: Complex Analysis
Hille: Analytic functions

On Monday 20.11 I lectured on pages 1-6 and will start on Stirling's series on Thursday.
Thursday 23.11: I lectured on pages 7-19. I will start on Theorem 7.3 page 20 on Monday. I handed out one page (Note CB/22.11.06) about Stirling's formula and a proof that the radius of convergence of Stirling's series is 0.

Week 48: On Monday 27.11 I finished the notes on the Gamma function and began to lecture on Riemann's zeta function. Notes about this function will be handed out on Nov. 30. There are several books on Riemann's zeta function:

H.M. Edwards, Riemann's zeta function, 1974.
A. Ivic, The Riemann Zeta Function, 1985.
E.C. Titchmarsh, The theory of the Riemann zeta-function, 1951

Thursday 30.11: I handed out notes on Riemann's zeta function-pages, 1-15. I proved the functional equation for zeta, but I still have to say something about Lemma 2.10 next week, where I will finish the notes handed out and begin to discuss Dirichlet series.

Week 49 On December 4 I finished the pages 1-15 and started on page 16 about Dirichlet series. I handed out pages 16-29 plus 2 pages with the graphs of M(x) and \sqrt{x}. Also one page with a poem about the Riemann conjecture. I almost finished Jensen's lemma 2.16.
On Thursday 7.12 I finished the notes on Dirichlet series. If you want more information about prime numbers and their distribution, look at
Paulo Ribenboim: The new boook of prime number records, Third edition 1996. He writes among other things: Von Mangoldts theorem that $\sum \mu(n)/n=0$ is equivalent to the Riemann hypothesis which is also equivalent to the following behaviour of Mertens function M(x), namely $M(x)=O(x^{1/2+\varepsilon},x\to\infty$ for any $\varepsilon >0$.
Concerning the prime number theorem I can recommend the article by P.T. Bateman and H.G. Diamond: A hundred years of prime numbers, Amer. Math. Monthly 103 (1996), 729--741.

Week 50 I handed out pages 1-13 of Chapter 3: Hypergeometric functions. I lectured on pages 1-7 and will start on Thursday 14.12 on Euler's Theorem 3.6. I also handed out the project catalogue.
You can find the many formulas about hypergeometric functions on the link Hypergeometric which contains 1260 formulas.
On 14.12 I handed out the remaining pages 14-32 on hypergeometric functions as well as 2 pages called: Another proof og Gauss' formula for F(a,b;c;1). I lectured on these pages and ended by formulating Fuch's theorem on page 15.

Week 51. Due to Christmas there will only be lectures on Monday 18.12. I lectured today on the pages 15-24 about hypergeometric functions. The remarks about singularities at infinity I will skip and start with Theorem 3.16 and the rest of the chapter when we resume refreshed on January 8.

Week 1. Thursday 4.1. This lecture is cancelled because of project work.

Week 2. On Monday Jan. 8 I finished the chapter on hypergeometric functions. I began to introduce elliptic functions by calculating the arc length of an ellipse. On Thursday I handed out the following notes on elliptic functions: (a) Printed pages: Elliptic functions (A front page, Chapter 1 from Akhiezer and a historical note) (b) Handwritten notes: Chapter 4, pages 1-23 and appendices 4.1 and 4.2.
I lectured on pages 1-17 and will start on Monday 15 on page 18.

Week 3. On Monday January 15 I finished Liouville's theorem about elliptic functions and introduced Weierstrass' p-function. I handed out notes pages 24-33a. On Thursday 18 I handed out notes 34-43 and I covered until page 40.

Week 4 On Monday 22 I handed out the last notes, pages 43-61 and I covered until page 48. The course will finish on Thursday 25. I covered the rest of the material.