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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Laplace transform of the psi function


Author: Atul Dixit
Journal: Proc. Amer. Math. Soc. 138 (2010), 593-603
MSC (2000): Primary 33B15
Published electronically: September 25, 2009
MathSciNet review: 2557176
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Abstract: An expression for the Laplace transform of the psi function

$\displaystyle L(a):=\int_{0}^{\infty}e^{-at}\psi(t+1) dt\\ $    

is derived using two different methods. It is then applied to evaluate the definite integral

$\displaystyle M(a)=\frac{4}{\pi} \int_{0}^{\infty}\frac{x^2 dx}{x^2+\ln^{2}(2e^{-a}\cos x)},\\ $    

for $ a>\ln 2$ and to resolve a conjecture posed by Olivier Oloa.


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Additional Information

Atul Dixit
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: aadixit2@illinois.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10157-0
PII: S 0002-9939(09)10157-0
Keywords: Laplace transform, gamma function, psi function.
Received by editor(s): October 28, 2008
Published electronically: September 25, 2009
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.