A Note on Some Integrals by Malmsten and Bierens de Haan

Abstract

Carl Johan Malmsten (1846) and David Beirens de Haan (1847) published work containing some interesting integrals. While no formal derivations of the integrals in his book Nouvelles Tables d'Intègrales Dèfines are available in current literature deriving and evaluating such formulae are useful in all aspects of science and engineering whenever such formulae are used. Formulae in the book of Bierens de Haan are used in connection with certain potential problems where there is the need to determine the vector potential of two parallel, infinitely long, tubular rectangular conductors carrying cur-rents in opposite directions. In this current work we supply formal derivations for some of these integrals along with deriving some special cases as new integrals in order to expand upon the book of Bierens de haan to aid in potential research where these formulae are applicable. Updating book of integrals is always a useful exercise as it keeps the volume accurate and more useful for potential readers and researchers. Formal derivations are also useful as they help in verifying the correctness of integrals in such volumes. The definite integral we derived in this work is given by <img src=image/13424330_01.gif> (1) in terms of the Lerch function, where the parameters a; k; m; and p are general complex numbers subject to their restrictions. This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.