On the integral \(\int ^{\infty}_{0}x^{\nu -1}(1+\beta x)^{- \lambda}\,\ln ^ mx\,dx\)
type: Item
instance of: scholarly article
title: On the integral \(\int ^{\infty}_{0}x^{\nu -1}(1+\beta x)^{- \lambda}\,\ln ^ mx\,dx\)
zbMATH Open document ID: 0591.65015
DOI: 10.1016/0377-0427(86)90070-1
author: K. S. Kölbig
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(M_ m(\beta,\nu,\lambda)\) denote the integral in the title, with \(| \arg \beta | <\pi\), \(0<Re \nu <Re \lambda\) and \(m=0,1,2,... \). The author proves that the function \(G_ m(\beta,\nu,\lambda)\), defined by \[ M_ m(\beta,\nu,\lambda)=\beta^{-\nu}\frac{\Gamma (\nu)\Gamma (\lambda -\nu)}{\Gamma (\lambda)}G_ m(\beta,\nu,l) \] satisfies a recurrence relation and is easily computed with a formula manipulation system such as REDUCE. A table of the resulting expressions for \(m=1(1)6\) is presented. Closed representations for \(M_ m(\beta,\nu,\lambda)\) for integer or half-integer values of \(\nu\) and \(\lambda\) are derived. Tables of explicit expressions for \(M_ m(\beta,\nu,\lambda)\) for small values of m, with \(\nu =n\), \(\nu =n+\), \(\lambda =\ell\), \(\lambda =\ell +\) and certain values of \(\beta\), comparison of results from the direct and recursive computation and an interesting discussion of some formulas related to the results of this paper are also given.
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 33E99
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 3949580
zbMATH Keywords: definite integrals
zbMATH Keywords: logarithmic integrals
zbMATH Keywords: integral tables
zbMATH Keywords: infinite series
zbMATH Keywords: formula manipulation
zbMATH Keywords: Stirling numbers
zbMATH Keywords: Bernoulli
zbMATH Keywords: numbers
zbMATH Keywords: Euler numbers
zbMATH Keywords: Bell polynomials
zbMATH Keywords: Mellin integrals
reviewed by: Luigi Gatteschi
describes a project that uses: REDUCE
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90070-1
OpenAlex ID: W235081597
cites work: Q5558293
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cites work: On the Integral � ∞ 0 e -μt t t ν- 1 log m t dt
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description: scientific article; zbMATH DE number 3949580