Extended Euler-Maclaurin summation formulas
type: Item
instance of: scholarly article
title: Extended Euler-Maclaurin summation formulas
zbMATH Open document ID: 0801.40003
DOI: 10.1007/BF00970967
author: Birutė Kryžienė
published in: Lithuanian Mathematical Journal
publication date: 1994-07-19T00:00:00Z
review text: The well-known Euler-Maclaurin summation formula may be viewed as an asymptotic formula for the sum \((1/n) \sum_{k=0}^{n-1} f((k+\theta)/n)\), \(0<\theta \leq 1\), for \(n\) tending to infinity. Usually, the function \(f\) is assumed to have continuous derivatives up to some order on the interval \([0,1]\). The paper, which is a revision and extension of earlier papers of Navot and the author, gives asymptotics in the case of \(f\) having some singularity at one endpoint of the interval (of the form \(x^{-\alpha}\) \(\alpha>0\), where \(\alpha \not\in \mathbb{N}\) or \(x^{-\alpha} \log x\) where \(\alpha\geq 0\), \(\alpha\not\in \mathbb{N}\)). The idea is the following: \(f\) is represented in a form \(f= f_ 1+h\) where the function \(h\) is smooth and \(f_ 1\) is a finite sum of elementary functions having singularities. Then the Euler-Maclaurin formula is applied to the functions \(f_ 1\) and \(h\) separately.
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 65B15
zbMATH DE Number: 578649
zbMATH Keywords: generalized Riemann zeta-function
zbMATH Keywords: Euler-Maclaurin summation formula
zbMATH Keywords: asymptotics
reviewed by: Jürgen Müller
MaRDI profile type: Publication
cites work: Convergence of distributions of extremal independent random variables
cites work: Q3209932
cites work: Q3794968
full work available at URL: https://doi.org/10.1007/bf00970967
OpenAlex ID: W2064354749
description: scientific article; zbMATH DE number 578649