Editing Lambert W function (section)

== History ==
Lambert first considered the related ''Lambert's Transcendental Equation'' in 1758,<ref>Lambert J. H., [http://www.kuttaka.org/~JHL/L1758c.pdf "Observationes variae in mathesin puram"], ''Acta Helveticae physico-mathematico-anatomico-botanico-medica'', Band III, 128–168, 1758.</ref> which led to an article by [[Leonhard Euler]] in 1783<ref>Euler, L. [http://math.dartmouth.edu/~euler/docs/originals/E532.pdf "De serie Lambertina Plurimisque eius insignibus proprietatibus"]. ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921.</ref> that discussed the special case of {{math|''we<sup>w</sup>''}}.

The equation Lambert considered was 
: <math>x = x^m + q.</math>

Euler transformed this equation into the form
: <math>x^a - x^b = (a - b) c x^{a + b}.</math>

Both authors derived a series solution for their equations.

Once Euler had solved this equation, he considered the case {{tmath|1=a = b}}. Taking limits, he derived the equation
: <math>\ln x = c x^a.</math>

He then put {{tmath|1=a = 1}} and obtained a convergent series solution for the resulting equation, expressing {{tmath|x}} in terms of&nbsp;{{tmath|c}}.

After taking derivatives with respect to {{tmath|x}} and some manipulation, the standard form of the Lambert function is obtained.

In 1993, it was reported that the Lambert {{tmath|W}} function provides an exact solution to the quantum-mechanical [[Delta potential#Double delta potential|double-well Dirac delta function model]] for equal charges<ref>{{cite journal |last1=Scott |first1=TC |last2=Babb |first2=JF |last3=Dalgarno |first3=A |last4=Morgan |first4=John D |title=The calculation of exchange forces: General results and specific models |journal=J. Chem. Phys. |date=Aug 15, 1993 |volume=99 |issue=4 |pages=2841–2854 |doi=10.1063/1.465193 |publisher=American Institute of Physics |bibcode=1993JChPh..99.2841S |language=English |issn=0021-9606}}</ref>—a fundamental problem in physics. Prompted by this, Rob Corless and developers of the [[Maple software|Maple]] [[computer algebra system]] realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."<ref name = "Corless" /><ref name="corless_maple">{{cite journal |first1=R. M. |last1=Corless |first2=G. H. |last2=Gonnet |first3=D. E. G. |last3=Hare |first4=D. J. |last4=Jeffrey |title=Lambert's {{tmath|W}} function in Maple |journal=The Maple Technical Newsletter |volume=9 |pages=12–22 |year=1993 |citeseerx = 10.1.1.33.2556 }}</ref>

Another example where this function is found is in [[Michaelis–Menten kinetics]].<ref>{{Cite book |last=Mező |first=István |url=https://www.taylorfrancis.com/books/mono/10.1201/9781003168102/lambert-function-istv%C3%A1n-mez%C5%91 |title=The Lambert W Function: Its Generalizations and Applications |year=2022 |doi=10.1201/9781003168102|isbn=9781003168102 |s2cid=247491347 }}</ref>

Although it was widely believed that the Lambert {{tmath|W}} function cannot be expressed in terms of elementary ([[Liouvillian function|Liouvillian]]) functions, the first published proof did not appear until 2008.<ref>{{cite journal |first1=Manuel |last1=Bronstein |first2=Robert M. |last2=Corless |first3=James H. |last3=Davenport |first4=D. J. |last4= Jeffrey |title=Algebraic properties of the Lambert {{tmath|W}} function from a result of Rosenlicht and of Liouville |journal=Integral Transforms and Special Functions |volume=19 |issue=10 |pages=709–712 |year=2008 |doi=10.1080/10652460802332342 |s2cid=120069437 |url=http://opus.bath.ac.uk/27004/1/Davenport_ITSF_19_10_709.pdf |archive-url=https://web.archive.org/web/20151211132056/http://opus.bath.ac.uk/27004/1/Davenport_ITSF_19_10_709.pdf |archive-date=2015-12-11 |url-status=live }}</ref>