Starts 15 Feb 2022 16:00

Ends 23 Mar 2022 15:30

Central European Time

SISSA/IGAP Lecture on "Standard and less standard asymptotic methods"

Starts 15 Feb 2022 16:00

Ends 23 Mar 2022 15:30

Central European Time

Online

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

**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.

**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.**

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