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The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

Author(s): Tewodros Amdeberhan; Olivier Espinosa; Victor H. Moll
Journal: Proc. Amer. Math. Soc. 136 (2008), 3211-3221.
MSC (2000): Primary 33B15
Posted: April 30, 2008
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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 F{í}sica, Universidad T{é}c. 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: 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
Posted: April 30, 2008
Additional Notes: The work of the third author was partially funded by NSF-DMS 0409968.
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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