The Lambert transform for small and large values of the transformation parameter
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
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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.
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ferr 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.
gold R. R. Goldberg, Inversions of generalized Lambert transforms, Duke Math. J. vol. 25, 1958, pp. 459–476.
goyal 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.
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milleri 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.
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pennington W. B. Pennington, Widder’s inversion formula for the Lambert transform, Duke Math., vol. 27, 1960, pp. 561–568.
rainai 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.
rainaii 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.
paley 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.
rodero J. Rodero Carrasco, The Lambert transform and an inversion formula, Rev. Acad. Ci. Madrid, vol. 53, 1959, pp. 727–736.
widder D. V. Widder, An inversion of the Lambert transform, Math. Mag, vol. 23, 1950, pp. 171–182.
wong R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989.
zayed A. I. Zayed, Handbook of Function and Generalized Function Transformations, CRC Press, New York, 1996.
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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
ORCID:
0000-0002-6050-9015
Email:
jl.lopez@unavarra.es
Keywords:
Lambert transform,
asymptotic expansions,
error bounds
Received by editor(s):
November 28, 2005
Published electronically:
August 9, 2006
Article copyright:
© Copyright 2006
Brown University