Editing Lambert W function (section)

== Representations ==
The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:<ref name = "Finch">{{cite book | last = Finch | first = S. R. | title = Mathematical constants | page = 450 | year = 2003 | publisher = Cambridge University Press }}</ref>
: <math>-\frac{\pi}{2}W_0(-x)=\int_0^\pi\frac{\sin\left(\tfrac32 t\right)-xe^{\cos t}\sin\left(\tfrac52 t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^2e^{2\cos t}}\sin\left(\tfrac12 t\right)\,dt \quad \text{for } |x| < \frac1{e}.
</math>

Another representation of the principal branch was found by Kalugin–Jeffrey–Corless:<ref>
{{cite journal
 | last1 = Kalugin | first1 = German A.
 | last2 = Jeffrey | first2 = David J.
 | last3 = Corless | first3 = Robert M.
 | doi = 10.1080/10652469.2011.640327
 | issue = 11
 | journal = Integral Transforms and Special Functions
 | mr = 2989751
 | pages = 817–829
 | title = Bernstein, Pick, Poisson and related integral expressions for Lambert {{mvar|W}}
 | url = https://www.uwo.ca/apmaths/faculty/jeffrey/pdfs/BersteinPick.pdf
 | volume = 23
 | year = 2012
}} See Theorem 3.4, p. 821 of published version (p. 5 of preprint).</ref> 
: <math>W_0(x)=\frac{1}{\pi}\int_0^\pi\ln\left(1+x\frac{\sin t}{t}e^{t\cot t}\right)dt.</math>

The following [[continued fraction]] representation also holds for the principal branch:<ref name = "Dubinov">
{{cite book
 | last1 = Dubinov| first1 = A. E.
 | last2 = Dubinova| first2 = I. D.
 | last3 = Saǐkov| first3 = S. K.
 | title = The Lambert ''W'' Function and Its Applications to Mathematical Problems of Physics (in Russian)
 | page = 53
 | year = 2006
 | publisher = RFNC-VNIIEF
}}</ref>
: <math>
W_0(x) = \cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{2+\cfrac{5x}{3+\cfrac{17x}{10+\cfrac{133x}{17+\cfrac{1927x}{190+\cfrac{13582711x}{94423+\ddots}}}}}}}}.
</math>
Also, if {{math|{{abs|''W''<sub>0</sub>(''x'')}} < 1}}:<ref>{{Cite book |last1=Robert M. |first1=Corless |last2=David J. |first2=Jeffrey |last3=Donald E. |first3=Knuth |title=Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97 |chapter=A sequence of series for the Lambert ''W'' function |year=1997 |pages=197–204 |doi=10.1145/258726.258783| isbn=978-0897918756 |s2cid=6274712 }}</ref>
: <math>W_0(x) = \cfrac{x}{\exp \cfrac{x}{\exp \cfrac{x}{\ddots}}}.</math>
In turn, if {{math|{{abs|''W''<sub>0</sub>(''x'')}} > 1}}, then
: <math>W_0(x) = \ln \cfrac{x}{\ln \cfrac{x}{\ln \cfrac{x}{\ddots}}}.</math>