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

Ferreira, Chelo; López, José

doi:10.1090/S0033-569X-06-01014-5

Authors: Chelo Ferreira and José L. López
Journal: Quart. Appl. Math. 64 (2006), 515-527
MSC (2000): Primary 41A60, 65R10; Secondary 33B15
DOI: https://doi.org/10.1090/S0033-569X-06-01014-5
Published electronically: August 9, 2006
MathSciNet review: 2259052
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive asymptotic expansions of the Lambert transform \[ \int _0^\infty xt(e^{xt}-1)^{-1}f(t)dt\] 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.

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
Chelo Ferreira and José L. López, An asymptotic expansion of the double gamma function, J. Approx. Theory 111 (2001), no. 2, 298–314. MR 1849551, DOI https://doi.org/10.1006/jath.2001.3578
José L. López and Chelo Ferreira, Asymptotic expansions of generalized Stieltjes transforms of algebraically decaying functions, Stud. Appl. Math. 108 (2002), no. 2, 187–215. MR 1883093, DOI https://doi.org/10.1111/1467-9590.01404
Richard R. Goldberg, Inversions of generalized Lambert transforms, Duke Math. J. 25 (1958), 459–476. MR 95390
S. P. Goyal and R. K. Laddha, On the generalized Riemann zeta functions and the generalized Lambert transform, Gaṇita Sandesh 11 (1997), no. 2, 99–108 (1998). MR 1641564
N. Hayek, B. J. González, and E. R. Negrin, The generalized Lambert transform on distributions of compact support, J. Math. Anal. Appl. 275 (2002), no. 2, 938–944. MR 1943789, DOI https://doi.org/10.1016/S0022-247X%2802%2900241-X
Eldon L. Miller, Summability of power series using the Lambert transform, Univ. Nac. Tucumán Rev. Ser. A 28 (1978), no. 1-2, 89–106 (1985). MR 839840
E. L. Miller, Convergence properties and an inversion formula for the Lambert transform, Rev. Técn. Fac. Ingr. Univ. Zulia 2 (1979), no. 1-2, 77–94 (English, with Spanish summary). MR 549310
E. R. Negrín, The Lambert transform on $\epsilon ’$. I, Portugal. Math. 50 (1993), no. 3, 359–363. MR 1244514
F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
W. B. Pennington, Widder’s inversion formula for the Lambert transform, Duke Math. J. 27 (1960), 561–568. MR 116187
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. 31 (2000), no. 1, 49–56. MR 1745770
R. K. Raina and H. M. Srivastava, Certain results associated with the generalized Riemann zeta functions, Rev. Técn. Fac. Ingr. Univ. Zulia 18 (1995), no. 3, 301–304 (English, with English and Spanish summaries). MR 1372687
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
Julián Rodero Carrasco, The Lambert transform and an inversion formula, Rev. Acad. Ci. Madrid 53 (1959), 727–736 (Spanish). MR 125417
D. V. Widder, An inversion of the Lambert transform, Math. Mag. 23 (1950), 171–182. MR 36864, DOI https://doi.org/10.2307/3029825
R. Wong, Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR 1016818
Ahmed I. Zayed, Handbook of function and generalized function transformations, Mathematical Sciences Reference Series, CRC Press, Boca Raton, FL, 1996. MR 1392476