SISSA/IGAP/SUSTech Lectures on "Standard and less standard asymptotic methods"

Zagier, Don (2022) SISSA/IGAP/SUSTech Lectures on "Standard and less standard asymptotic methods". [Video] (Unpublished)

[thumbnail of Part 1] Video (Part 1)
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[thumbnail of Part 3] Video (Part 3)
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[thumbnail of Part 4] Video (Part 4)
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[thumbnail of Part 5] Video (Part 5)
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[thumbnail of Part 6] Video (Part 6)
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[thumbnail of Part 7] Video (Part 7)
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[thumbnail of Part 8] Video (Part 8)
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[thumbnail of Part 10] Video (Part 10)
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Abstract

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.

Item Type: Video
Divisions: Research > Talks
Depositing User: This Admin
Date Deposited: 15 Sep 2022 08:28
Last Modified: 15 Sep 2022 08:34
URI: https://archive.mpim-bonn.mpg.de/id/eprint/4779

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