The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa
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- by Tewodros Amdeberhan, Olivier Espinosa and Victor H. Moll PDF
- Proc. Amer. Math. Soc. 136 (2008), 3211-3221 Request permission
Abstract:
The definite integral \begin{equation*} M(a):= \frac {4}{\pi } \int _{0}^{\pi /2} \frac {x^{2} dx } {x^{2} + \ln ^{2}( 2 e^{-a} \cos x ) }\end{equation*} is related to the Laplace transform of the digamma function \begin{equation*} L(a) := \int _{0}^{\infty } e^{-a s} \psi (s+1) ds, \end{equation*} 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.References
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Additional Information
- Tewodros Amdeberhan
- Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
- MR Author ID: 260444
- 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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3211-3221
- MSC (2000): Primary 33B15
- DOI: https://doi.org/10.1090/S0002-9939-08-09300-3
- MathSciNet review: 2407086