Scientific Calendar Event

 Description 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 Register in advance for this meeting: https://unesco-org.zoom.us/meeting/register/tJUvcO2qqDgtGtMYJCHfEtIForecwtoJIK4r You will receive a confirmation email containing information about joining the meeting. Description: 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
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