Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

The Laplace transform of the psi function

Author(s): Atul Dixit
Journal: Proc. Amer. Math. Soc. 138 (2010), 593-603.
MSC (2000): Primary 33B15
Posted: September 25, 2009
MathSciNet review: 2557176
Retrieve article in: PDF

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:

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)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33B15

Retrieve articles in all Journals with MSC (2000): 33B15

Additional Information:

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

DOI: 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
Posted: September 25, 2009
Communicated by: Peter A. Clarkson
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|