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Lambert W function

(Definition)

Lambert's function is the inverse of the function $f: \mathbbmss{C}\to \mathbbmss{C}$ given by . That is, is the complex valued function that satisfies

$\begin{displaymath}W(x) e^{W(x)} = x,\end{displaymath}$

for all $x \in \mathbbmss{C}$ . In practice the definition of

requires a branch cut, which is usually taken along the negative real axis. Lambert's W function is sometimes also called product log function.

This function allow us to solve the functional equation

$\begin{displaymath}g(x)^{g(x)}=x\end{displaymath}$

since

$\begin{displaymath}g(x)=e^{W(\ln(x))}.\end{displaymath}$

References

A site with good information on Lambert's W function is Corless' page “On the Lambert W Function”

"Lambert W function" is owned by drini. [ owner history (1) ]

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Other names:

product log

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Cross-references: functional equation, real axis, negative, cut, branch, complex, inverse, function
There is 1 reference to this entry.

This is version 5 of Lambert W function, born on 2002-05-28, modified 2005-02-28.
Object id is 2957, canonical name is LambertWFunction.
Accessed 11669 times total.

Classification:

AMS MSC:

33B30 (Special functions :: Elementary classical functions :: Higher logarithm functions)

Pending Errata and Addenda

None.[ View all 3 ]

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