Editing Lambert W function (section)

{{Short description|Multivalued function in mathematics}}
{{Use American English|date = March 2019}}
{{DISPLAYTITLE:Lambert ''W'' function}}
[[File:The product logarithm Lambert W function plotted in the complex plane from -2-2i to 2+2i.svg|upright=1.3|alt=The product logarithm Lambert W function plotted in the complex plane from −2 − 2''i'' to 2 + 2''i''|thumb|The product logarithm Lambert W function plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[File:Mplwp lambert W branches.svg|thumb|upright=1.3|The graph of {{math|1=''y'' = ''W''(''x'')}} for real {{math|''x'' < 6}} and {{math|''y'' > −4}}. The upper branch (blue) with {{math|''y'' ≥ −1}} is the graph of the function {{math|''W''<sub>0</sub>}} (principal branch), the lower branch (magenta) with {{math|''y'' ≤ −1}} is the graph of the function {{math|''W''<sub>−1</sub>}}. The minimum value of {{math|''x''}} is at {{math|{{mset|−1/''e'', −1}}}}]]

In [[mathematics]], the '''Lambert {{mvar|W}} function''', also called the '''omega function''' or '''product logarithm''',<ref>
{{citation
 | last = Lehtonen | first = Jussi
 | editor-last = Rees | editor-first = Mark
 | date = April 2016
 | doi = 10.1111/2041-210x.12568
 | issue = 9
 | journal = Methods in Ecology and Evolution
 | pages = 1110–1118
 | title = The Lambert W function in ecological and evolutionary models
 | volume = 7
 | s2cid = 124111881 | doi-access = free
 | bibcode = 2016MEcEv...7.1110L
}}</ref> is a [[multivalued function]], namely the [[Branch point|branches]] of the [[converse relation]] of the function {{math|1=''f''(''w'') = ''we''<sup>''w''</sup>}}, where {{mvar|w}} is any [[complex number]] and {{math|''e''<sup>''w''</sup>}} is the [[exponential function]]. The function is named after [[Johann Heinrich Lambert|Johann Lambert]], who considered a related problem in 1758. Building on Lambert's work, [[Leonhard Euler]] described the {{mvar|W}} function per se in 1783.{{Citation needed|date=October 2024}}

For each integer {{math|''k''}} there is one branch, denoted by {{math|''W''<sub>''k''</sub>(''z'')}}, which is a complex-valued function of one complex argument. {{math|''W''<sub>0</sub>}} is known as the [[principal branch]]. These functions have the following property: if {{math|''z''}} and {{math|''w''}} are any complex numbers, then
: <math>w e^{w} = z</math>
holds if and only if
: <math>w=W_k(z) \ \ \text{ for some integer } k.</math>

When dealing with real numbers only, the two branches {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}} suffice: for real numbers {{math|''x''}} and {{math|''y''}} the equation
: <math>y e^{y} = x</math>
can be solved for {{math|''y''}} only if {{math|''x'' ≥ −{{sfrac|1|''e''}}}}; yields {{math|1=''y'' = ''W''<sub>0</sub>(''x'')}} if {{math|''x'' ≥ 0}} and the two values {{math|1=''y'' = ''W''<sub>0</sub>(''x'')}} and {{math|1=''y'' = ''W''<sub>−1</sub>(''x'')}} if {{math|−{{sfrac|1|''e''}} ≤ ''x'' < 0}}.

The Lambert {{mvar|W}} function's branches cannot be expressed in terms of [[elementary function]]s.<ref>
{{citation
 | last = Chow | first = Timothy Y.
 | issue = 5
 | journal = [[American Mathematical Monthly]]
 | pages = 440–448
 | title = What is a closed-form number?
 | volume = 106
 | year = 1999
 | mr = 1699262 | arxiv = math/9805045 | jstor = 2589148 | doi = 10.2307/2589148
}}.</ref> It is useful in [[combinatorics]], for instance, in the enumeration of [[tree graph|trees]]. It can be used to solve various equations involving exponentials (e.g. the maxima of the [[Planck's law|Planck]], [[Bose–Einstein distribution|Bose–Einstein]], and [[Fermi–Dirac distribution|Fermi–Dirac]] distributions) and also occurs in the solution of [[delay differential equation]]s, such as {{math|1=''y''′(''t'') = ''a'' ''y''(''t'' − 1)}}. In [[biochemistry]], and in particular [[enzyme kinetics]], an opened-form solution for the time-course kinetics analysis of [[Michaelis–Menten kinetics]] is described in terms of the Lambert {{mvar|W}} function.

[[File:Cplot Lambert W.png|thumb|upright=1.3|Main branch of the Lambert {{mvar|W}} function in the complex plane, plotted with [[domain coloring]]. Note the [[branch cut]] along the negative real axis, ending at {{math|−{{sfrac|1|''e''}}}}.]]
[[File:Lambert-modulus-HSV.png|thumb|upright=1.3|The modulus of the principal branch of the Lambert {{mvar|W}} function, colored according to {{math|[[argument (complex analysis)|arg]] ''W''(''z'')}}]]