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Difference between revisions of "Lambert transform"

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The integral transform
 
The integral transform
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$$
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F(x) = \int_0^\infty \frac{t a(t)}{e^{xt}-1} dt \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573601.png" /></td> </tr></table>
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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
 
+
$$
The Lambert transform is the continuous analogue of the [[Lambert series|Lambert series]] (under the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573603.png" />). The following inversion formula holds: Suppose that
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a(t) \in L(0,\infty)
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573604.png" /></td> </tr></table>
 
 
 
 
and that
 
and that
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$$
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\lim_{t \rightarrow +0} a(t) t^{1-\delta} = 0,\ \ \delta > 0 \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573605.png" /></td> </tr></table>
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If also $\tau > 0$ and if the function $a(t)$ is continuous at $t = \tau$, then one has
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$$
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\tau a(\tau) = \lim_{k\rightarrow\infty} \frac{(-1)^k}{k!} \left({\frac{k}{\tau}}\right)^{k+1} \sum_{n=1}^\infty \mu(n) n^k F^{(k)}\left({ \frac{nk}{\tau} }\right) \,,
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$$
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where $\mu(n)$ is the [[Möbius function]].
  
If also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573606.png" /> and if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573607.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573608.png" />, then one has
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====References====
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<table>
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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">[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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</table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l0573609.png" /></td> </tr></table>
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{{TEX|done}}
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057360/l05736010.png" /> is the [[Möbius function|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>
 

Revision as of 20:49, 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
[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
How to Cite This Entry:
Lambert transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_transform&oldid=19166
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