On the integral \(\int ^{\infty}_{0}x^{\nu -1}(1+\beta x)^{- \lambda}\,\ln ^ mx\,dx\)
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.
scientific article; zbMATH DE number 3949580
scientific article