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The Laplace transform of the psi function


Author: Atul Dixit
Journal: Proc. Amer. Math. Soc. 138 (2010), 593-603
MSC (2000): Primary 33B15
DOI: https://doi.org/10.1090/S0002-9939-09-10157-0
Published electronically: September 25, 2009
MathSciNet review: 2557176
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. T. Amdeberhan, O. Espinosa and V.H. Moll, The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa, Proc. Amer. Math. Soc., 136 (2008) 3211-3221. MR 2407086 (2009d:33005)
  • 2. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642 (29:4914)
  • 3. J. B Conway, Functions of One Complex Variable, 2nd ed., Springer-Verlag, New York-Berlin, 1978. MR 503901 (80c:30003)
  • 4. O. Espinosa and V.H. Moll, On some integrals involving the Hurwitz zeta function. Part I, The Ramanujan Journal, 6 (2002) 159-188. MR 1908196 (2003f:11127)
  • 5. M.L. Glasser and D. Manna, On the Laplace transform of the psi function, ``Tapas in Experimental Mathematics'' (T. Amdeberhan and V. Moll, eds.), Contemporary Mathematics, vol. 457, Amer. Math. Soc., Providence, RI, 2008, pp. 205-214. MR 2427673
  • 6. I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, Boston, MA, 1994. MR 1243179 (94g:00008)
  • 7. A.P. Guinand, Some formulae for the Riemann zeta-function, J. London Math. Soc., 22 (1947) 14-18. MR 0022928 (9:279g)
  • 8. O. Oloa, Some Euler-type integrals and a new rational series for Euler's constant, ``Tapas in Experimental Mathematics'' (T. Amdeberhan and V. Moll, eds.), Contemporary Mathematics, vol. 457, Amer. Math. Soc., Providence, RI, 2008, 253-264. MR 2427676
  • 9. E.T. Whittaker and G.N Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, London, 1927. MR 0178117 (31:2375)

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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: https://doi.org/10.1090/S0002-9939-09-10157-0
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.

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