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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 digamma function: An integral due to Glasser, Manna and Oloa


Authors: Tewodros Amdeberhan, Olivier Espinosa and Victor H. Moll
Journal: Proc. Amer. Math. Soc. 136 (2008), 3211-3221
MSC (2000): Primary 33B15
Published electronically: April 30, 2008
MathSciNet review: 2407086
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Abstract | References | Similar Articles | Additional Information

Abstract: The definite integral

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

is related to the Laplace transform of the digamma function

$\displaystyle L(a) := \int_{0}^{\infty} e^{-a s} \psi(s+1) \, ds, $

by $ M(a) = L(a) + \gamma/a$ when $ a > \ln 2$. Certain analytic expressions for $ M(a)$ in the complementary range, $ 0 < a \leq \ln 2$, are also provided.


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Additional Information

Tewodros Amdeberhan
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: tamdeber@tulane.edu

Olivier Espinosa
Affiliation: Departmento de Física, Universidad Téc. Federico Santa María, Valparaiso, Chile
Email: olivier.espinosa@usm.cl

Victor H. Moll
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: vhm@math.tulane.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09300-3
PII: S 0002-9939(08)09300-3
Keywords: Laplace transform, digamma function
Received by editor(s): July 23, 2007
Published electronically: April 30, 2008
Additional Notes: The work of the third author was partially funded by NSF-DMS 0409968.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.