Note on a Stieltjes Transform in terms of the Lerch Function

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \...

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Bibliographic Details
Published inEuropean journal of pure and applied mathematics Vol. 14; no. 3; pp. 723 - 736
Main Authors Reynolds, Robert, Stauffer, Allan
Format Journal Article
LanguageEnglish
Published 01.07.2021
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ISSN1307-5543
1307-5543
DOI10.29020/nybg.ejpam.v14i3.3991

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Summary:In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.
ISSN:1307-5543
1307-5543
DOI:10.29020/nybg.ejpam.v14i3.3991