Please does someone know a good description of how Euler did derive his summation formula? Thank you!
Asked
Active
Viewed 930 times
6
Martin Sleziak
- 53,687
thatseries
- 61
-
See also the references in http://en.wikipedia.org/wiki/Euler–Maclaurin_formula – Dietrich Burde Apr 27 '14 at 11:44
1 Answers
6
Most of Euler's work is available at "The Euler Archive".
The papers of interest in this case (according to Ferraro's "Euler's derivation of the Euler–Maclaurin summation formula") are :
- $1732$-$33$ (E25) "Methodus generalis summandi progressiones" translated here.
- $1736$ (E47) "Inventio summae cuiusque seriei ex dato termino generali" translated here.
Further references :
Ferraro's article "Some Aspects of Euler's Theory of Series - Inexplicable Functions and the Euler–Maclaurin Summation Formula" details Euler's different derivations.
Hairer and Wanner's book "Analysis by its History" provides a short Euler derivation.
Pengelley's "Dances between continuous and discrete" considers Euler's derivations too.
Apostol's "An Elementary View of Euler's Summation Formula" may clarify things.
Lampret's derivation may be of interest too.
Raymond Manzoni
- 43,021
- 5
- 86
- 140
-
Oh cool. I was reading chapter five of Ramanujan's second notebook which led me to Bruce Berndt's discussion of said chapter which led me to Raymond Ayoub's paper 'Euler and the Zeta Function' which led me to check out Euler's derivation of the Euler-Maclaurin formula which led me to search MSE which led me to this nice list. The Ferraro text in particular looks a valuable resource, I'll definitely have to check it out. – Beauty Is Truth Mar 15 '17 at 06:07
-
@BeautyIsTruth: Glad my list pleased you! Concerning Euler's work you may like Sandifer's simple introductions from the first link. Back to Ramanujan's scanned versions of his original notebooks were made available not so long ago and may be found for example here (nsrBook1.pdf to nsrBook3.pdf). Great surfing, – Raymond Manzoni Mar 15 '17 at 23:16