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The Lambert transform for small and large values of the transformation parameter

Author(s): Chelo Ferreira; José L. López
Journal: Quart. Appl. Math. 64 (2006), 515-527.
MSC (2000): Primary 41A60, 65R10; Secondary 33B15
Posted: August 9, 2006
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Abstract: We derive asymptotic expansions of the Lambert transform

$\displaystyle \int_0^\infty xt(e^{xt}-1)^{-1}f(t)dt$

of a locally integrable function $ f(t)$ for small and large $ x$. All the expansions are accompanied by error bounds for the remainder at any order of the approximation.

References:

1.
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, New York, 1966. MR 0208797 (34:8606)

2.
C. Ferreira and J. L. López, An Asymptotic Expansion of the Double Gamma Function, J. of Approx. Theory. vol. 111, 2001, pp. 298-314. MR 1849551 (2002g:41049)

3.
C. Ferreira and J. L. López, Asymptotic expansions of generalized Stieltjes transforms of algebraically decaying functions, Stud. Appl. Math. vol. 108, 2002, pp. 187-215. MR 1883093 (2003c:44007)

4.
R. R. Goldberg, Inversions of generalized Lambert transforms, Duke Math. J. vol. 25, 1958, pp. 459-476. MR 0095390 (20:1893)

5.
S. P. Goyal and R. K. Laddha, On the generalized Riemann zeta functions and the generalized Lambert transform, Gadnita Sandesh. vol. 11, n. 2, 1997, pp. 99-108. MR 1641564 (2000b:11106)

6.
N. Hayek, B. J. González and E. R. Negrin, The generalized Lambert transform on distributions of compact support, J. Math. Anal. Appl. vol. 275, 2002, pp. 938-944. MR 1943789 (2003k:46051)

7.
E. L. Miller, Summability of power series using the Lambert transform, Rev. Univ. Nac. Tucumán, Series A vol. 28, n. 1,2, 1978, pp. 89-106. MR 0839840 (87g:40003)

8.
E. L. Miller, Convergence properties and an inversion formula for the Lambert transform, Rev. Tech. Ingr. Univ. Zulia vol. 2, n. 1,2, 1979, pp. 77-94. MR 0549310 (80j:44006)

9.
E. R. Negrín, The Lambert transform on $ \epsilon^\prime$, Portugal Math, vol. 50, n. 3, 1993, pp. 359-363. MR 1244514 (94i:44004)

10.
F.W.J. Olver, Special Functions: Asymptotics and Special Functions, Academic Press, New York, 1974. MR 0435697 (55:8655)

11.
W. B. Pennington, Widder's inversion formula for the Lambert transform, Duke Math., vol. 27, 1960, pp. 561-568. MR 0116187 (22:6982)

12.
R. K. Raina and T. S. Nahar, A note on a certain class of functions related to Hurwitz zeta function and Lambert transform, Tamkang J. Math, vol. 31, n.1, 2000, pp. 49-56. MR 1745770 (2000m:11079)

13.
R. K. Raina and H. M. Srivastava, Certain results associated with the generalized Riemann zeta functions, Rev. Tecn. Fac. Ingr. Univ. Zulia, vol. 18, n.3, 1995, pp. 301-304. MR 1372687 (96m:11074)

14.
R.E.A.C. Paley and N. Wiener, Fourier transforms in the complex domain, Reprint of the 1934 original. Amer. Math. Soc. Coll. Publ., vol. 19, 1987. MR 1451142 (98a:01023)

15.
J. Rodero Carrasco, The Lambert transform and an inversion formula, Rev. Acad. Ci. Madrid, vol. 53, 1959, pp. 727-736. MR 0125417 (23:A2720)

16.
D. V. Widder, An inversion of the Lambert transform, Math. Mag, vol. 23, 1950, pp. 171-182. MR 0036864 (12:175c)

17.
R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989. MR 1016818 (90j:41061)

18.
A. I. Zayed, Handbook of Function and Generalized Function Transformations, CRC Press, New York, 1996. MR 1392476 (97h:44001)

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Additional Information:

Chelo Ferreira
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Spain
Email: cferrei@unizar.es

José L. López
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, Spain
Email: jl.lopez@unavarra.es

PII: S0033-569X-06-01014-5
Keywords: Lambert transform, asymptotic expansions, error bounds
Received by editor(s): November 28, 2005
Posted: August 9, 2006
Copyright of article: Copyright 2006, Brown University


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