Starts 15 Feb 2022 16:00
Ends 23 Mar 2022 15:30
Central European Time
Leonardo Building - Budinich Lecture Hall
Lecturer: Don Zagier
Course Type: SISSA PhD Course
Academic Year: 2021-2022
Period: mid February to mid March 2022
Research Group: SISSA-Geometry and Mathematical Physics
Duration: 20 h

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In every branch of mathematics, one is sometimes confronted with the problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of  formation of a sequence of numbers {A_n} of which one knows, for  instance, the first couple of hundred values.  The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much  less so, with lots of examples.   Here are three typical examples: 1. The slowly convergent sum  sum_{j=0}^\infty (\binom{j+4/3}{j})^{-4/3}   arose in the work of a colleague.  Evaluate it to 250 decimal digits. 2.  Expand the infinite sum  \sum_{n=0}^\infty (1-q)(1-q^2)...(1-q^n) as  \sum A_n (1-q)^n, with first coefficients 1, 1, 2, 5, 15, 53, ... Show numerically that  A_n  is asymptotic to  n! * a * n^b * c  for some real constants a, b and c, evaluate all three to high precision, and recognize their exact values. 3.  The infinite series  H(x) = \sum_{k=1}^\infty sin(x/k)/k  converges for every complex number x.  Compute this series to high accuracy when x is a large real number, so that the series is highly oscillatory.

Next lecture:
12 Wed 23-Mar 14.00 - 15.30

These will be hybrid courses. All are very welcome to join either online or in person (if provided with a green pass). Venue: Budinich Lecture Hall (ICTP Leonardo Da Vinci Building), for those wishing to attend in person.