View source for Lambert transform

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====
<table>
<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>
<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>
</table>

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