View source for Lambert transform

The integral transform

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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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and that

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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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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>