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Difference between revisions of "Lambert transform"

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<TR><TD valign="top">[1]</TD> <TD valign="top">  D.V. Widder,  "An inversion of the Lambert transform"  ''Math. Mag.'' , '''23'''  (1950)  pp. 171–182</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  D.V. Widder,  "An inversion of the Lambert transform"  ''Math. Mag.'' , '''23'''  (1950)  pp. 171–182 {{DOI|10.2307/3029825}}  {{ZBL|0036.35302}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms"  ''Progress in Math.'' , '''4'''  (1969)  pp. 1–85  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms"  ''Progress in Math.'' , '''4'''  (1969)  pp. 1–85  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82 {{ZBL|0197.37903}}</TD></TR>
 
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Latest revision as of 21:25, 4 October 2017

The integral transform $$ F(x) = \int_0^\infty \frac{t a(t)}{e^{xt}-1} dt \ . $$

The Lambert transform is the continuous analogue of the Lambert series (under the correspondence $t a(t) \leftrightarrow a_n$, $e^x \leftrightarrow 1/t$). The following inversion formula holds: Suppose that $$ a(t) \in L(0,\infty) $$ and that $$ \lim_{t \rightarrow +0} a(t) t^{1-\delta} = 0,\ \ \delta > 0 \ . $$

If also $\tau > 0$ and if the function $a(t)$ is continuous at $t = \tau$, then one has $$ \tau a(\tau) = \lim_{k\rightarrow\infty} \frac{(-1)^k}{k!} \left({\frac{k}{\tau}}\right)^{k+1} \sum_{n=1}^\infty \mu(n) n^k F^{(k)}\left({ \frac{nk}{\tau} }\right) \,, $$ where $\mu(n)$ is the Möbius function.

References

[1] D.V. Widder, "An inversion of the Lambert transform" Math. Mag. , 23 (1950) pp. 171–182 DOI 10.2307/3029825 Zbl 0036.35302
[2] V.A. Ditkin, A.P. Prudnikov, "Integral transforms" Progress in Math. , 4 (1969) pp. 1–85 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82 Zbl 0197.37903
How to Cite This Entry:
Lambert transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_transform&oldid=42008
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article