View source for Lambert W function

{{Use American English|date = March 2019}}
{{Short description|Multivalued "function"}}
{{DISPLAYTITLE:Lambert ''W'' function}}
[[Image:Lambert-w.svg|thumb|288px|right|The graph of {{math|''W''(''x'')}} for {{math|''W'' > −4}} and {{math|''x'' < 6}}. The upper branch with {{math|''W'' ≥ −1}} is the function {{math|''W''<sub>0</sub>}} (principal branch), the lower branch with {{math|''W'' ≤ −1}} is the function {{math|''W''<sub>−1</sub>}}.]]

In [[mathematics]], the '''Lambert {{mvar|W}} function''', also called the '''omega function''' or '''product logarithm''', is a set of [[function (mathematics)|functions]], namely the [[Branch point|branches]] of the [[inverse relation]] of the function {{math|''f''(''z'') {{=}} ''ze''<sup>''z''</sup>}}, where {{math|''e''<sup>''z''</sup>}} is the [[exponential function]], and {{mvar|z}} is any [[complex number]] (x+iy). In other words, if {{math|''w'' {{=}} ''W''(''z'')}} is any branch of the Lambert {{mvar|W}} function then

:<math> z = f(w) = we^w.</math>

This is not quite the same as the similar-looking statement in reverse order
:<math>w = f^{-1}\left(z\right) = W\left(we^w\right),</math>
which requires that the correct branch of {{mvar|W}} be chosen on the right-hand side in order to ensure equality.

With that caveat, by substituting {{math|''w'' {{=}} ''W''(''z'')}} into the first equation above, we get the defining equation for the {{mvar|W}} function (and for the {{mvar|W}} relation in general):

:<math>z = W(z) e^{W(z)}</math>

for any complex number {{math|''z''}}.

The relation {{mvar|W}} is [[multivalued function|multivalued]] (except at 0).  If we restrict attention to real-valued {{mvar|W}}, the complex variable {{mvar|z}} is then replaced by the real variable {{mvar|x}}, and the relation is defined only for {{math|''x'' ≥ −1/''e''}} and is double-valued on the interval {{math|(−1/''e'', 0)}}. The additional constraint {{math|''W'' ≥ −1}} defines a single-valued function {{math|''W''<sub>0</sub>(''x'')}}.  We have {{math|''W''<sub>0</sub>(0) {{=}} 0}} and {{math|''W''<sub>0</sub>(−1/''e'') {{=}} −1}}.  Meanwhile, the lower branch has {{math|''W'' ≤ −1}} and is denoted {{math|''W''<sub>−1</sub>(''x'')}}.  It decreases from {{math|''W''<sub>−1</sub>(−1/''e'') {{=}} −1}} to {{math|''W''<sub>−1</sub>(0<sup>−</sup>) {{=}} −∞}}.

It can be extended to the function {{math|''z'' {{=}} ''xa<sup>x</sup>''}} using the identity

:<math>x = \frac{W(z\ln{a})}{\ln{a}}.</math>

The Lambert {{mvar|W}} relation cannot be expressed in terms of [[elementary function]]s.<ref>{{citation
 | last = Chow | first = Timothy Y.
 | doi = 10.2307/2589148
 | issue = 5
 | journal = [[American Mathematical Monthly]]
 | mr = 1699262
 | pages = 440–448
 | title = What is a closed-form number?
 | volume = 106
 | year = 1999| arxiv = math/9805045
 | jstor = 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|''y''′(''t'') {{=}} ''a'' ''y''(''t'' − 1)}}. In [[biochemistry]], and in particular [[enzyme kinetics]], a closed-form solution for the time-course [[Kinetics analysis.co.in|kinetics analysis]] of [[Michaelis–Menten kinetics]] is described in terms of the Lambert {{mvar|W}} function.

:[[Image:Product Log.jpg|thumb|288px|Main branch of the Lambert {{mvar|W}} function in the complex plane.  Note the [[branch cut]] along the negative real axis, ending at {{math|−1/''e''}}. In this picture, the hue of a point {{mvar|z}} is determined by the [[argument (complex analysis)|argument]] of {{math|''W''(''z'')}}, and the brightness by the [[absolute value]] of {{math|''W''(''z'')}}.]]

[[Image:Lambert-modulus-HSV.png|thumb|300px|The modulus of the principal branch of the Lambert {{mvar|W}} function, colored according to {{math|[[argument (complex analysis)|arg]] ''W''(''z'')}}]]

==Terminology==
[[File:Diagram of the real branches of the Lambert W function.png|thumb|The two main branches {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}}.]]
The Lambert {{mvar|W}} function is named after [[Johann Heinrich Lambert]]. The main branch {{math|''W''<sub>0</sub>}} is denoted {{mvar|Wp}} in the [[Digital Library of Mathematical Functions]], and the branch {{math|''W''<sub>−1</sub>}} is denoted {{mvar|Wm}} there.

The notation convention chosen here (with {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}}) follows the canonical reference on the Lambert {{mvar|W}} function by Corless, Gonnet, Hare, Jeffrey and [[Donald Knuth|Knuth]].<ref name = "Corless">{{cite journal
  | last1 = Corless  | first1 = R. M.
  | last2 = Gonnet | first2 = G. H.
  | last3 = Hare | first3 = D. E. G.
  | last4 = Jeffrey | first4 = D. J.
  | last5 = Knuth | first5 = D. E.
  | author5-link = Donald Knuth
  | title = On the Lambert ''W'' function
  | journal = Advances in Computational Mathematics
  | volume = 5
  | pages = 329–359
  | year = 1996
  | url = http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps | format = PostScript
  | doi = 10.1007/BF02124750
 | arxiv = 1809.07369
  }}</ref>

==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 function 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 {{math|1=''a'' = ''b''}}. Taking limits he derived the equation
: <math>\ln x = c x^a.</math>

He then put {{math|1=''a'' = 1}} and obtained a convergent series solution.

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

In 1993, when it was reported that the Lambert {{mvar|W}} function provides an exact solution to the quantum-mechanical [[Delta potential#Double delta potential|double-well Dirac delta function model]] for equal charges—a fundamental problem in physics—Corless and developers of the [[Maple software|Maple]] computer algebra system made a library search and found that this function was ubiquitous in nature.<ref name = "Corless" /><ref name="corless_maple">{{cite journal |first=R. M. |last=Corless |first2=G. H. |last2=Gonnet |first3=D. E. G. |last3=Hare |first4=D. J. |last4=Jeffrey |title=Lambert's W function in Maple |journal=The Maple Technical Newsletter |volume=9 |issue= |pages=12–22 |year=1993 |citeseerx = 10.1.1.33.2556 }}</ref>

Another example where this function is found is in [[Michaelis–Menten kinetics]].

==Calculus==

===Derivative===
By [[implicit differentiation]], one can show that all branches of {{mvar|W}} satisfy the [[ordinary differential equation|differential equation]]

:<math>z(1 + W) \frac{dW}{dz} = W \quad \text{for } z \neq -\frac{1}{e}.</math>

({{mvar|W}} is not [[Differentiable function|differentiable]] for {{math|''z'' {{=}} −1/''e''}}.) As a consequence, we get the following formula for the derivative of ''W'':

:<math>\frac{dW}{dz} = \frac{W(z)}{z(1 + W(z))} \quad \text{for } z \not\in \left\{0, -\frac{1}{e}\right\}.</math>

Using the identity {{math|''e''<sup>''W''(''z'')</sup> {{=}} ''z'' / ''W''(''z'')}}, we get the following equivalent formula:
:<math>\frac{dW}{dz} = \frac{1}{z + e^{W(z)}} \quad \text{for } z \neq -\frac{1}{e}.</math>

===Antiderivative===
The function {{math|''W''(''x'')}}, and many expressions involving {{math|''W''(''x'')}}, can be [[integral|integrated]] using the [[substitution rule|substitution]] {{math|''w'' {{=}} ''W''(''x'')}}, i.e. {{math|''x'' {{=}} ''we''<sup>''w''</sup>}}:

:<math> \begin{align}
\int W(x)\,dx &= x W(x) - x + e^{W(x)} + C\\
& = x \left( W(x) - 1 + \frac{1}{W(x)} \right) + C.
\end{align}</math>
(The last equation is more common in the literature but does not hold at {{math|''x'' {{=}} 0}}).  One consequence of this (using the fact that {{math|''W''(''e'') {{=}} 1}}) is the identity
:<math>\int_{0}^{e} W(x)\,dx = e - 1.</math>

==Asymptotic expansions==
The [[Taylor series]] of {{math|''W''<sub>0</sub>}} around 0 can be found using the [[Lagrange inversion theorem]] and is given by

:<math>W_0(x)=\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}x^n =x-x^2+\tfrac{3}{2}x^3-\tfrac{8}{3}x^4+\tfrac{125}{24}x^5-\cdots.</math>

The [[radius of convergence]] is {{math|1/''e''}}, as may be seen by the [[ratio test]]. The function defined by this series can be extended to a [[holomorphic function]] defined on all complex numbers with a [[branch cut]] along the [[interval (mathematics)|interval]] {{math|(−∞, −1/''e'']}}; this holomorphic function defines the [[principal branch]] of the Lambert {{mvar|W}} function.

For large values of {{mvar|x}}, {{math|''W''<sub>0</sub>}} is asymptotic to
:<math>\begin{align}
W_0(x) &= L_1 - L_2 + \frac{L_2}{L_1} + \frac{L_2\left(-2 + L_2\right)}{2L_1^2} + \frac{L_2\left(6 - 9L_2 + 2L_2^2\right)}{6L_1^3} + \frac{L_2\left(-12 + 36L_2 - 22L_2^2 + 3L_2^3\right)}{12L_1^4} + \cdots \\[5pt]
&= L_1 - L_2 + \sum_{l=0}^\infty \sum_{m=1}^\infty \frac{(-1)^l \left[ \begin{smallmatrix} l + m \\ l + 1 \end{smallmatrix} \right]}{m!} L_1^{-l-m} L_2^m,
\end{align}</math>
where {{math|''L''<sub>1</sub> {{=}} ln ''x''}}, {{math|''L''<sub>2</sub> {{=}} ln ln ''x''}}, and {{math|<big><big>[</big></big>{{su|p=''l'' + ''m''|b=''l'' + 1|a=c}}<big><big>]</big></big>}} is a non-negative [[Stirling numbers of the first kind|Stirling number of the first kind]].<ref name = "Corless" /> Keeping only the first two terms of the expansion,
:<math>W_0(x) = \ln x - \ln \ln x + o(1).</math>

The other real branch, {{math|''W''<sub>−1</sub>}}, defined in the interval {{math|[−1/''e'', 0)}}, has an approximation of the same form as {{mvar|x}} approaches zero, with in this case {{math|''L''<sub>1</sub> {{=}} ln(−''x'')}} and {{math|''L''<sub>2</sub> {{=}} ln(−ln(−''x''))}}.<ref name = "Corless" />

It is shown<ref>A. Hoorfar, M. Hassani, [https://www.emis.de/journals/JIPAM/article983.html?sid=983 Inequalities on the Lambert ''W'' Function and Hyperpower Function], JIPAM, volume 9, issue 2, article 51. 2008.</ref> that the following bound holds (the upper bound only for {{math|''x'' ≥ ''e''}}):
:<math>\ln x -\ln \ln x + \frac{\ln \ln x}{2\ln x} \le W_0(x) \le \ln x - \ln\ln x + \frac{e}{e - 1} \frac{\ln \ln x}{\ln x}.</math>

In 2013 it was proven<ref name = "Chatzigeorgiou">{{cite journal
  | last1 = Chatzigeorgiou| first1 = I.
  | title = Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation
  | journal = IEEE Communications Letters
  | volume = 17
  | issue = 8
  | pages = 1505–1508
  | year = 2013
  | arxiv = 1601.04895| doi = 10.1109/LCOMM.2013.070113.130972
 }}</ref> that the branch {{math|''W''<sub>−1</sub>}} can be bounded as follows:
:<math>-1 - \sqrt{2u} - u < W_{-1}\left(-e^{-u-1}\right) < -1 - \sqrt{2u} - \tfrac{2}{3}u \quad \text{for } u > 0.</math>

===Integer and complex powers===
Integer powers of {{math|''W''<sub>0</sub>}} also admit simple [[Taylor series|Taylor]] (or [[Laurent series|Laurent]]) series expansions at zero:

:<math>
W_0(x)^2 = \sum_{n=2}^\infty \frac{-2\left(-n\right)^{n-3}}{(n - 2)!} x^n = x^2 - 2x^3 + 4x^4 - \tfrac{25}{3}x^5 + 18x^6 - \cdots.
</math>

More generally, for {{math|''r'' ∈ '''ℤ'''}}, the [[Formal power series#The Lagrange inversion formula|Lagrange inversion formula]] gives

:<math>
W_0(x)^r = \sum_{n=r}^\infty \frac{-r\left(-n\right)^{n - r - 1}}{(n - r)!} x^n,
</math>
which is, in general, a Laurent series of order {{mvar|r}}. Equivalently, the latter can be written in the form of a Taylor expansion of powers of {{math|''W''<sub>0</sub>(''x'') / ''x''}}:

:<math>
\left(\frac{W_0(x)}{x}\right)^r = e^{-r W_0(x)} = \sum_{n=0}^\infty \frac{r\left(n + r\right)^{n - 1}}{n!} \left(-x\right)^n,
</math>
which holds for any {{math|''r'' ∈ '''ℂ'''}} and {{math|{{abs|''x''}} < 1/''e''}}.

==Identities==
A few identities follow from the definition:
:<math>\begin{align}
W(x e^x) &= x & \text{for } x &\geq 0,\\
W_0(x e^x) &= x & \text{for } x &\geq -1,\\
W_{-1}(x e^x) &= x & \text{for } x &\leq -1.
\end{align}</math>
Note that, since {{math|''f''(''x'') {{=}} ''xe<sup>x</sup>''}} is not [[injective]], it does not always hold that {{math|''W''(''f''(''x'')) {{=}} ''x''}}, much like with the [[inverse trigonometric functions]]. For fixed {{math|''x'' < 0}} and {{math|''x'' ≠ −1}}, the equation {{math|''xe<sup>x</sup>'' {{=}} ''ye<sup>y</sup>''}} has two solutions in {{mvar|y}}, one of which is of course {{math|''y'' {{=}} ''x''}}. Then, for {{math|''i'' {{=}} 0}} and {{math|''x'' < −1}}, as well as for {{math|''i'' {{=}} −1}} and {{math|''x'' ∈ (−1, 0)}}, {{math|''y'' {{=}} ''W<sub>i</sub>''(''xe<sup>x</sup>'')}} is the other solution.

Some other identities:<ref>{{cite web | url=http://functions.wolfram.com/ElementaryFunctions/ProductLog/17/01/0001/ | title=Lambert function: Identities (formula 01.31.17.0001)}}</ref>
:<math>
\begin{align}
& W(x) \cdot e^{W(x)} = x, \quad\text{therefore:}\\[5pt]
& e^{W(x)} = \frac{x}{W(x)}, \qquad e^{-W(x)} = \frac{W(x)}{x}, \qquad e^{n W(x)} = \left(\frac{x}{W(x)}\right)^n.
\end{align}
</math>

:<math>\ln W(x) = \ln x - W(x) \quad \text{for } x > 0.</math><ref>{{cite web | url=http://mathworld.wolfram.com/LambertW-Function.html | title=Lambert W-Function}}</ref>

: <math>W\left(a \ln a\right) = \ln a \quad \text{for } a\in \mathbb{R}. </math>

: <math>
\begin{align}
& W(x) = \ln \frac{x}{W(x)} &&\text{for } x \geq -\frac1e, \\[5pt]
& W\left( \frac{nx^n}{W\left(x\right)^{n-1}} \right) = n W(x) &&\text{for } n, x > 0
\end{align}
</math>

::(which can be extended to other {{mvar|n}} and {{mvar|x}} if the right branch is chosen).

:<math>W(x) + W(y) = W\left(x y \left(\frac{1}{W(x)} + \frac{1}{W(y)}\right)\right) \quad \text{for } x, y > 0.</math>

From inverting {{math|''f''(ln ''x'')}}:
:<math>\begin{align}
W(x \ln x) &= \ln x \\
           &= W(x) + \ln W(x) \quad\text{for } x > 0.
\end{align}</math>

Substituting {{math|−ln ''x''}} in the definition:
:<math>\begin{align}
W_0\left(-\frac{\ln x}{x}\right) &= -\ln x &\text{for } 0 &< x \leq e,\\[5pt]
W_{-1}\left(-\frac{\ln x}{x}\right) &= -\ln x &\text{for } x &> e.
\end{align}</math>

With Euler's iterated exponential {{math|''h''(''x'')}}:
:<math>\begin{align}h(x) & = e^{-W(-\ln x)}\\
                         & = \frac{W(-\ln x)}{-\ln x} \quad \text{for } x \neq 1.
\end{align}</math>

==Special values==
For any nonzero [[algebraic number]] {{mvar|x}}, {{math|''W''(''x'')}} is a [[transcendental number]]. Indeed, if {{math|''W''(''x'')}} is zero, then {{mvar|x}} must be zero as well, and if {{math|''W''(''x'')}} is nonzero and algebraic, then by the [[Lindemann–Weierstrass theorem]], {{math|''[[e (mathematical constant)|e]]''<sup>''W''(''x'')</sup>}} must be transcendental, implying that {{math|''x'' {{=}} ''W''(''x'')''e''<sup>''W''(''x'')</sup>}} must also be transcendental.

: <math>W\left(-\frac{\pi}{2}\right) = \frac{i\pi}{2}.</math>

: <math>W\left(-\frac{1}{e}\right) = -1.</math>

: <math>W\left(-\frac{\ln a}{a}\right)= -\ln a  \quad \left(\frac{1}{e}\le a\le e\right).</math>

: <math>W\left(2 \ln 2 \right) = \ln 2.</math>

: <math>W(0) = 0.</math>

: <math>W(1) = \Omega = \left(\int_{-\infty}^{\infty} \frac{dt}{\left(e^t-t\right)^2 + \pi^2}\right)^{-1} - 1\approx 0.56714329\ldots</math> (the [[Omega constant]]).

: <math>W(1) = e^{-W(1)} = \ln\left(\frac{1}{W(1)}\right) = -\ln W(1).</math>

: <math>W(e) = 1.</math>

: <math>W(e^{1+e}) = e.</math>

: <math>W(-1) \approx -0.31813+1.33723i.</math>

: <math>W'(0) = 1.</math>

==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(-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>

On the wider domain {{math|−1/''e'' ≤ ''x'' ≤ ''e''}}, the considerably simpler representation is found by Mező:<ref>{{cite web |last1=István |first1=Mező |title=An integral representation for the principal branch of Lambert the ''W'' function |url=https://sites.google.com/site/istvanmezo81/others |accessdate=7 November 2017}}</ref>
:<math>
W(x) = \frac{1}{\pi} \operatorname{Re} \int_0^\pi \ln\left(\frac{e^{e^{it}} - xe^{-it}}{e^{e^{it}} - xe^{it}}\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(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''(''z'')}} < 1}}:<ref>{{Cite book |last1=Robert M. |first1=Corless |last2=David J. |first2=Jeffrey |last3=Donald E. |first3=Knuth |title=A sequence of series for the Lambert ''W'' function |journal=Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation |year=1997 |pages=197–204 |doi=10.1145/258726.258783|isbn=978-0897918756 }}</ref>
:<math>
W(x) = \cfrac{x}{\exp \cfrac{x}{\exp \cfrac{x}{\ddots}}}.
</math>
In turn, if {{math|{{abs|''W''(''z'')}} > e}}, then
:<math>
W(x) = \ln \cfrac{x}{\ln \cfrac{x}{\ln \cfrac{x}{\ddots}}}.
</math>

==Branches and range==
[[File:Lambert W Range.pdf|thumb|right|The range of the {{mvar|W}} function, showing all branches. The orange curves are the images of the imaginary axis. The black are the images of the real axis (the [[quadratrix of Hippias]]). The purple curves is the image of a small circle around the branch point {{math|''z'' {{=}} 0}}; the red curves are the images of a small circle around the point {{math|''z'' {{=}} −1/''e''}}.]]
There are countably many branches of the {{mvar|W}} function, denoted by {{math|''W<sub>k</sub>''(''z'')}}, for integer {{mvar|k}}; {{math|''W''<sub>0</sub>(''z'')}} being the principal branch.  The branch point for the principal branch is at {{math|''z'' {{=}} −1/''e''}}, with a branch cut that extends to {{math|−∞}} along the negative real axis.  This branch cut separates the principal branch from the two branches {{math|''W''<sub>−1</sub>}} and {{math|''W''<sub>1</sub>}}.  In all other branches, there is a branch point at {{math|''z'' {{=}} 0}} and a branch cut along the whole negative real axis.

The range of the entire function is the complex plane.  The image of the real axis coincides with the [[quadratrix of Hippias]], the parametric curve {{math|''w'' {{=}} −''t'' cot ''t'' + ''it''}}.

==Other formulas==

===Definite integrals===
There are several useful definite integral formulas involving the {{mvar|W}} function, including the following:

:<math>
\begin{align}
& \int_0^\pi W\left( 2\cot^2x \right)\sec^2 x\,dx = 4\sqrt{\pi}. \\[5pt]
& \int_0^\infty \frac{W(x)}{x\sqrt{x}}\,dx = 2\sqrt{2\pi}. \\[5pt]
& \int_0^\infty W\left(\frac{1}{x^2}\right)\,dx = \sqrt{2\pi}.
\end{align}
</math>

The first identity can be found by writing the [[Gaussian integral]] in [[polar coordinates]].

The second identity can be derived by making the substitution {{math|''u'' {{=}} ''W''(''x'')}}, which gives

:<math>\begin{align}
x & =ue^u, \\[5pt]
\frac{dx}{du} & =(u+1)e^u.
\end{align}</math>

Thus

:<math>\begin{align}
\int_0^\infty \frac{W(x)}{x\sqrt{x}}\,dx &=\int_0^\infty \frac{u}{ue^{u}\sqrt{ue^{u}}}(u+1)e^u \, du \\[5pt]
&=\int_0^\infty \frac{u+1}{\sqrt{ue^u}}du \\[5pt]
&=\int_0^\infty \frac{u+1}{\sqrt{u}}\frac{1}{\sqrt{e^u}}du\\[5pt]
&=\int_0^\infty u^\tfrac12 e^{-\frac{u}{2}}du+\int_0^\infty u^{-\tfrac12} e^{-\frac{u}{2}}du\\[5pt]
&=2\int_0^\infty (2w)^\tfrac12 e^{-w} \, dw+2\int_0^\infty (2w)^{-\tfrac12} e^{-w} \, dw && \quad (u =2w) \\[5pt]
&=2\sqrt{2}\int_0^\infty w^\tfrac12 e^{-w} \, dw + \sqrt{2} \int_0^\infty w^{-\tfrac12} e^{-w} \, dw \\[5pt]
&=2\sqrt{2} \cdot \Gamma \left (\tfrac32 \right )+\sqrt{2} \cdot \Gamma \left (\tfrac12 \right ) \\[5pt]
&=2\sqrt{2} \left (\tfrac12\sqrt{\pi} \right )+\sqrt{2}\left(\sqrt{\pi}\right) \\[5pt]
&=2\sqrt{2\pi}.
\end{align}</math>

The third identity may be derived from the second by making the substitution {{math|''u'' {{=}} ''x''<sup>−2</sup>}} and the first can also be derived from the third by the substitution {{math|''z'' {{=}} (tan ''x'') / {{sqrt|2}}}}.

Except for {{mvar|z}} along the branch cut {{math|(−∞, −1/''e'']}} (where the integral does not converge), the principal branch of the Lambert {{mvar|W}} function can be computed by the following integral:<ref>{{cite web|title=The Lambert ''W'' Function|url=http://www.orcca.on.ca/LambertW/|publisher=Ontario Research Centre for Computer Algebra}}</ref>

:<math>\begin{align}
W(z)&=\frac{z}{2\pi}\int_{-\pi}^\pi\frac{\left(1-\nu\cot\nu\right)^2+\nu^2}{z+\nu\csc\nu e^{-\nu\cot\nu}} \, d\nu \\[5pt]
&= \frac{z}{\pi} \int_0^\pi \frac{\left(1-\nu\cot\nu\right)^2+\nu^2}{z + \nu \csc\nu e^{-\nu\cot\nu}} \, d\nu,
\end{align}</math>

where the two integral expressions are equivalent due to the symmetry of the integrand.

===Indefinite integrals===
:<math>\begin{align}
&\int \frac{W(x)}{x}dx=\tfrac{1}{2}\left(1+W(x)\right)^2+C. \\[5pt]
&\int W\left(A e^{Bx}\right)dx=\frac{1}{2B}\left(1+W\left(A e^{Bx}\right)\right)^2+C.
\end{align}</math>

==Applications==
===Viscous flows===
Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:
:<math>H(x)= 1 + W \left((H(0) -1) e^{(H(0)-1)-\frac{x}{L}}\right),</math>
where {{math|''H''(''x'')}} is the debris flow height, {{mvar|x}} is the channel downstream position, {{mvar|L}} is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.

In [[pipe flow]], the Lambert W function is part of the explicit formulation of the [[Colebrook equation]] for finding the [[Darcy friction factor]]. This factor is used to determine the pressure drop through a straight run of pipe when the flow is [[turbulent]].<ref name="AAMore">{{cite journal | title = Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes | author = More, A. A. | journal =  Chemical Engineering Science | volume = 61 | pages = 5515–5519 | year = 2006 | issue = 16 | doi = 10.1016/j.ces.2006.04.003}}</ref>

===Neuroimaging===
The Lambert {{mvar|W}} function was employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain voxel, to the corresponding blood oxygenation level dependent (BOLD) signal.<ref>{{cite journal |last1=Sotero |first1=Roberto C. |last2=Iturria-Medina |first2=Yasser |title=From Blood oxygenation level dependent (BOLD) signals to brain temperature maps |journal=Bull Math Biol |volume=73 |issue=11 |pages=2731–47 |year=2011 |doi=10.1007/s11538-011-9645-5 |pmid=21409512|url=http://precedings.nature.com/documents/4772/version/1 |type=Submitted manuscript }}</ref>

===Chemical engineering===
The Lambert {{mvar|W}} function was employed in the field of chemical engineering for modelling the porous electrode film thickness in a [[glassy carbon]] based [[supercapacitor]] for electrochemical energy storage. The Lambert {{mvar|W}} function turned out to be the exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.<ref>{{cite journal |last1=Braun |first1=Artur |last2=Wokaun |first2=Alexander |last3=Hermanns |first3=Heinz-Guenter |title=Analytical Solution to a Growth Problem with Two Moving Boundaries |journal=Appl Math Model |volume=27 |issue=1 |pages=47–52 |year=2003 |doi=10.1016/S0307-904X(02)00085-9}}</ref><ref>{{cite journal |last1=Braun |first1=Artur |last2=Baertsch |first2=Martin |last3=Schnyder |first3=Bernhard |last4=Koetz |first4=Ruediger |title=A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model. |journal=Chem Eng Sci |volume=55 |issue=22 |pages=5273–5282 |year=2000 |doi=10.1016/S0009-2509(00)00143-3}}</ref>

===Materials science===
The Lambert {{mvar|W}} function was employed in the field of [[Epitaxy|epitaxial film growth]] for the determination of the critical [[dislocation]] onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert {{mvar|W}} for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert {{mvar|W}} turns it in an explicit equation for analytical handling with ease.<ref>{{cite journal |last1=Braun |first1=Artur |last2=Briggs |first2=Keith M. |last3=Boeni |first3=Peter |title=Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films |journal=J Cryst Growth |volume=241 |issue=1–2 |pages=231–234 |year=2003 |doi=10.1016/S0022-0248(02)00941-7 |bibcode=2002JCrGr.241..231B}}</ref>

===Porous media===
The Lambert {{mvar|W}} function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the lighter fluid.<ref>{{cite journal|last1=Colla|first1=Pietro|year=2014|title=A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium|journal=PROCEEDINGS,Thirty-Eighth Workshop on Geothermal Reservoir Engineering,Stanford University|volume=SGP-TR-202}}([https://pangea.stanford.edu/ERE/pdf/IGAstandard/SGW/2014/Colla.pdf])</ref>

===Bernoulli numbers and Todd genus===
The equation (linked with the generating functions of [[Bernoulli number]]s and [[Genus of a multiplicative sequence|Todd genus]]):

:<math> Y = \frac{X}{1-e^X}</math>

can be solved by means of the two real branches {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}}:

:<math> X(Y) = \begin{cases}
W_{-1}\left( Y e^Y\right) - W_0\left( Y e^Y\right) = Y - W_0\left( Y e^Y\right) &\text{for }Y < -1,\\
W_0\left( Y e^Y\right) - W_{-1}\left( Y e^Y\right) = Y - W_{-1}\left(Y e^Y\right) &\text{for }-1 < Y < 0.
\end{cases}</math>

This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.<ref>[https://web.archive.org/web/20150212084155/http://www.apmaths.uwo.ca/~djeffrey/Offprints/SYNASC2014.pdf D. J. Jeffrey and J. E. Jankowski, "Branch differences and Lambert ''W''"]</ref>

===Statistics===
The centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence (also called the Jeffreys divergence) has a closed form using the Lambert {{mvar|W}} function.<ref>[http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6509930 F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"]</ref>

===Exact solutions of the Schrödinger equation===
The Lambert {{mvar|W}} function appears in a quantum-mechanical potential (see [[Lambert W step potential|Lambert {{mvar|W}} step potential]]) which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the [[inverse square root potential]] – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
:<math> V = \frac{V_0}{1+W \left(e^{-\frac{x}{\sigma}}\right)}.</math>
A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to<ref>[https://arxiv.org/abs/1509.00846 A.M. Ishkhanyan, "The Lambert ''W'' barrier – an exactly solvable confluent hypergeometric potential"].</ref>
:<math> z = W \left(e^{-\frac{x}{\sigma}}\right).</math>

===Exact solutions of the Einstein vacuum equations===
In the [[Schwarzschild metric]] solution of the Einstein vacuum equations, the {{mvar|W}} function is needed to go from the [[Eddington-Finkelstein coordinates]] to the Schwarzschild coordinates.  For this reason, it also appears in the construction of the [[Kruskal–Szekeres coordinates]].

===Resonances of the delta-shell potential===
The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert {{mvar|W}} function.<ref>{{cite journal |first=R. |last=de la Madrid |year=2017 |title=Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential |journal=Nucl. Phys. A |volume=962 |pages=24–45 |doi=10.1016/j.nuclphysa.2017.03.006 |arxiv=1704.00047 |bibcode=2017NuPhA.962...24D }}</ref>

===Thermodynamic equilibrium===
If a reaction involves reactants and products having [[heat capacity|heat capacities]] that are constant with temperature then the equilibrium constant {{mvar|K}} obeys

:<math>\ln K=\frac{a}{T}+b+c\ln T</math>

for some constants {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. When {{mvar|c}} (equal to {{math|Δ''C<sub>p</sub>'' / ''R''}}) is not zero we can find the value or values of {{mvar|T}} where {{mvar|K}} equals a given value as follows, where we use {{mvar|L}} for {{math|ln ''T''}}.

:<math>\begin{align}
-a&=(b-\ln K)T+cT\ln T\\
&=(b-\ln K)e^L+cLe^L\\[5pt]
-\frac{a}{c}&=\left(\frac{b-\ln K}{c}+L\right)e^L\\[5pt]
-\frac{a}{c}e^\frac{b-\ln K}{c}&=\left(L+\frac{b-\ln K}{c}\right)e^{L+\frac{b-\ln K}{c}}\\[5pt]
L&=W\left(-\frac{a}{c}e^\frac{b-\ln K}{c}\right)+\frac{\ln K-b}{c}\\[5pt]
T&=\exp\left(W\left(-\frac{a}{c}e^\frac{b-\ln K}{c}\right)+\frac{\ln K-b}{c}\right).
\end{align}</math>

If {{mvar|a}} and {{mvar|c}} have the same sign there will be either two solutions or none (or one if the argument of {{mvar|W}} is exactly {{math|−1/''e''}}). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.

===AdS/CFT correspondence===
The classical finite-size corrections to the dispersion relations of [[giant magnon]]s, single spikes and [[GKP string]]s can be expressed in terms of the Lambert {{mvar|W}} function.<ref>{{cite journal |first=Emmanuel |last=Floratos |first2=George |last2=Georgiou |first3=Georgios |last3=Linardopoulos |year=2014|title=Large-Spin Expansions of GKP Strings|journal=JHEP |volume=03|issue=3 |pages=0180|arxiv=1311.5800|doi=10.1007/JHEP03(2014)018}}</ref><ref>{{cite journal |first=Emmanuel |last=Floratos |first2=Georgios |last2=Linardopoulos |year=2015|title=Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes|journal=Nucl. Phys. B |volume=897|pages=229–275|arxiv=1406.0796|doi= 10.1016/j.nuclphysb.2015.05.021}}</ref>

==Generalizations==
The standard Lambert {{mvar|W}} function expresses exact solutions to ''transcendental algebraic'' equations (in {{mvar|x}}) of the form:

{{NumBlk|:|<math> e^{-c x} = a_0 (x-r)</math>|{{EquationRef|1}}}}

where {{math|''a''<sub>0</sub>}}, {{mvar|c}} and {{mvar|r}} are real constants.  The solution is
:<math> x = r + \frac{1}{c} W\left( \frac{c\,e^{-c r}}{a_0} \right). </math>
Generalizations of the Lambert {{mvar|W}} function<ref>{{cite journal |first=T. C. |last=Scott |first2=R. B. |last2=Mann |year=2006 |title=General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert ''W'' Function |journal=AAECC (Applicable Algebra in Engineering, Communication and Computing) |volume=17 |issue=1 |pages=41–47 |doi=10.1007/s00200-006-0196-1 |arxiv=math-ph/0607011 |last3=Martinez Ii |first3=Roberto E. }}</ref><ref>{{cite journal |first=T. C. |last=Scott |first2=G. |last2=Fee |first3=J.| last3=Grotendorst|year=2013 |title=Asymptotic series of Generalized Lambert W Function |journal=SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation) |volume=47 |issue=185 |pages=75–83|doi=10.1145/2576802.2576804|url=http://www.sigsam.org/cca/issues/issue185.html}}</ref><ref>{{cite journal |first=T. C. |last=Scott |first2=G. |last2=Fee |first3=J.| last3=Grotendorst|first4=W.Z. | last4=Zhang|year=2014|title=Numerics of the Generalized Lambert W Function|journal=SIGSAM |volume=48 |issue=1/2|pages=42–56|doi=10.1145/2644288.2644298|url=http://www.sigsam.org/cca/issues/issue188.html}}</ref> include:

*An application to [[general relativity]] and [[quantum mechanics]] ([[Quantum gravity#The dilaton|quantum gravity]]) in lower dimensions, in fact a previously unknown ''link'' (unknown prior to 2007<ref>{{cite journal |first=P. S. |last=Farrugia |first2=R. B. |last2=Mann |first3=T. C. |last3=Scott |year=2007 |title=''N''-body Gravity and the Schrödinger Equation |journal=Class. Quantum Grav. |volume=24 |issue=18 |pages=4647–4659 |doi=10.1088/0264-9381/24/18/006 |arxiv=gr-qc/0611144 |bibcode=2007CQGra..24.4647F }}</ref>) between these two areas,  where the right-hand side of ({{EquationNote|1}}) is now a quadratic polynomial in ''x'':
{{NumBlk|::|<math>e^{-c x} = a_0 \left(x-r_1 \right) \left(x-r_2 \right),</math>|{{EquationRef|2}}}}

:where {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function has a single argument {{mvar|x}} but the terms like {{math|''r<sub>i</sub>''}} and {{math|''a''<sub>0</sub>}} are parameters of that function.  In this respect, the generalization resembles the [[hypergeometric]] function and the [[Meijer G-function|Meijer {{mvar|G}} function]] but it belongs to a different ''class'' of functions. When {{math|''r''<sub>1</sub> {{=}} ''r''<sub>2</sub>}}, both sides of ({{EquationNote|2}}) can be factored and reduced to ({{EquationNote|1}}) and thus the solution reduces to that of the standard {{mvar|W}} function.  Equation ({{EquationNote|2}}) expresses the equation governing the [[dilaton]] field, from which is derived the metric of the [[R=T model|{{math|''R'' {{=}} ''T''}}]] or ''lineal'' two-body gravity problem in 1&nbsp;+&nbsp;1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical [[Delta potential#Double Delta Potential|double-well Dirac delta function model]] for ''unequal'' charges in one dimension.

*Analytical solutions of the eigenenergies of a special case of the quantum mechanical [[Euler's three-body problem|three-body problem]], namely the (three-dimensional) [[hydrogen molecule-ion]].<ref>{{cite journal |first=T. C. |last=Scott |first2=M. |last2=Aubert-Frécon |first3=J. |last3=Grotendorst |year=2006 |title=New Approach for the Electronic Energies of the Hydrogen Molecular Ion |journal=Chem. Phys. |volume=324 |issue=2–3 |pages=323–338 |doi=10.1016/j.chemphys.2005.10.031 |arxiv=physics/0607081 |bibcode=2006CP....324..323S |citeseerx=10.1.1.261.9067 }}</ref> Here the right-hand side of ({{EquationNote|1}}) (or ({{EquationNote|2}})) is now a ratio of infinite order polynomials in {{mvar|x}}:

{{NumBlk|::|<math> e^{-c x} = a_0 \frac{\displaystyle \prod_{i=1}^\infty (x-r_i )}{\displaystyle \prod_{i=1}^\infty (x-s_i)}</math>|{{EquationRef|3}}}}

:where {{math|''r''<sub>''i''</sub>}} and {{math|''s''<sub>''i''</sub>}} are distinct real constants and {{mvar|x}} is a function of the eigenenergy and the internuclear distance {{mvar|R}}. Equation ({{EquationNote|3}}) with its specialized cases expressed in ({{EquationNote|1}}) and ({{EquationNote|2}}) is related to a large class of [[delay differential equation]]s. [[G. H. Hardy]]'s notion of a "false derivative" provides exact multiple roots to special cases of ({{EquationNote|3}}).<ref>{{cite journal |first=Aude |last=Maignan |first2=T. C. |last2=Scott |year=2016|title=Fleshing out the Generalized Lambert ''W'' Function|journal=SIGSAM |volume=50 |issue=2|pages=45–60|doi=10.1145/2992274.2992275}}</ref>

Applications of the Lambert {{mvar|W}} function in fundamental physical problems are not exhausted even for the standard case expressed in ({{EquationNote|1}}) as seen recently in the area of [[atomic, molecular, and optical physics]].<ref>{{cite journal |first=T. C. |last=Scott |first2=A. |last2=Lüchow |first3=D. |last3=Bressanini |first4=J. D. III |last4=Morgan |year=2007 |title=The Nodal Surfaces of Helium Atom Eigenfunctions |journal=[[Physical Review|Phys. Rev. A]] |volume=75 |issue=6 |pages=060101 |doi=10.1103/PhysRevA.75.060101 |bibcode=2007PhRvA..75f0101S |url=https://irinsubria.uninsubria.it/bitstream/11383/1679348/1/2007-scott-pra2007.pdf }}</ref>

==Plots==
<gallery caption="Plots of the Lambert W function on the complex plane">
Image:LambertWRe.png|{{math|''z'' {{=}} Re(''W''<sub>0</sub>(''x'' + ''iy''))}}
Image:LambertWIm.png|{{math|''z'' {{=}} Im(''W''<sub>0</sub>(''x'' + ''iy''))}}
Image:LambertWAbs.png|{{math|''z'' {{=}} {{abs|''W''<sub>0</sub>(''x'' + ''iy'')}}}}
Image:LambertWAll.png|Superimposition of the previous three plots
</gallery>

==Numerical evaluation==
The {{mvar|W}} function may be approximated using [[Newton's method]], with successive approximations to {{math|''w'' {{=}} ''W''(''z'')}} (so {{math|''z'' {{=}} ''we<sup>w</sup>''}}) being
:<math>w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}+w_j e^{w_j}}.</math>

The {{mvar|W}} function may also be approximated using [[Halley's method]],
:<math>
w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}\left(w_j+1\right)-\dfrac{\left(w_j+2\right)\left(w_je^{w_j}-z\right)}{2w_j+2}}
</math>

given in Corless et al.<ref name="Corless"/> to compute {{mvar|W}}.

==Software==
The Lambert {{mvar|W}} function is implemented as
[http://www.maplesoft.com/support/help/Maple/view.aspx?path=LambertW <tt>'''LambertW'''</tt> in Maple], <tt>'''lambertw'''</tt> in [[PARI/GP|GP]] (and <tt>'''glambertW'''</tt> in [[PARI/GP|PARI]]), <tt>'''lambertw'''</tt> in [[Matlab]],<ref>[http://www.mathworks.com.au/help/toolbox/symbolic/lambertw.html lambertw – MATLAB]</ref> also <tt>'''lambertw'''</tt> in octave with the <tt>specfun</tt> package, as <tt>'''lambert_w'''</tt> in Maxima,<ref>[http://maxima.sourceforge.net Maxima, a Computer Algebra System]</ref> as <tt>'''ProductLog'''</tt> (with a silent alias <tt>'''LambertW'''</tt>) in [[Mathematica]],<ref>[http://reference.wolfram.com/mathematica/ref/ProductLog.html ProductLog at WolframAlpha]</ref> as <tt>'''lambertw'''</tt> in Python [[scipy]]'s special function package,<ref>{{cite web | url=http://docs.scipy.org/doc/scipy-0.16.0/reference/generated/scipy.special.lambertw.html | title=Scipy.special.lambertw — SciPy v0.16.1 Reference Guide}}</ref> as <tt>'''LambertW'''</tt> in Perl's <tt>ntheory</tt> module,<ref>[https://metacpan.org/pod/ntheory ntheory at MetaCPAN]</ref> and as <tt>'''gsl_sf_lambert_W0'''</tt>, <tt>'''gsl_sf_lambert_Wm1'''</tt> functions in the [https://www.gnu.org/software/gsl/manual/html_node/Lambert-W-Functions.html special functions] section of the [https://www.gnu.org/software/gsl/ GNU Scientific Library] (GSL). In the [https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/lambert_w.html Boost C++ libraries], the calls are <tt>'''lambert_w0'''</tt>, <tt>'''lambert_wm1'''</tt>, <tt>'''lambert_w0_prime'''</tt>, and <tt>'''lambert_wm1_prime'''</tt>. In R, the Lambert {{mvar|W}} function is implemented as the <tt>'''lambertW0'''</tt> and <tt>'''lambertWm1'''</tt> functions in the <tt>lamW</tt> package.<ref>{{Citation|last=Adler|first=Avraham|title=lamW: Lambert ''W'' Function|date=2017-04-24|url=https://cran.r-project.org/web/packages/lamW/index.html|accessdate=2017-12-19}}</ref>

A C++ code for all the branches of the complex Lambert {{mvar|W}} function is available on the homepage of István Mező.<ref>[https://sites.google.com/site/istvanmezo81/others The webpage of István Mező]</ref>

==See also==
* [[Wright Omega function]]
* Lambert's [[trinomial|trinomial equation]]
* [[Lagrange inversion theorem#Lambert W function|Lagrange inversion theorem]]
* [[Experimental mathematics]]
* [[Holstein–Herring method]]
* [[R=T model|{{math|''R'' {{=}} ''T''}} model]]
* [[Ross' π lemma|Ross' {{mvar|π}} lemma]]

==Notes==
{{reflist|30em}}

==References==
* {{Cite journal | last1=Corless | first1=R. | last2=Gonnet | first2=G. | last3=Hare | first3=D. | last4=Jeffrey | first4=D. | last5=Knuth | first5=Donald | author5-link=Donald Knuth | title=On the Lambert ''W'' function | url=http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf | year=1996 | journal=Advances in Computational Mathematics | issn=1019-7168 | volume=5 | pages=329–359 | doi=10.1007/BF02124750 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
* {{cite journal |url=http://www.istia.univ-angers.fr/~chapeau/papers/lambertw.pdf | last1 = Chapeau-Blondeau | first1 = F. | last2 = Monir | first2 = A. |title=Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2 |journal=IEEE Trans. Signal Process. |volume=50 |issue=9 |year=2002}}
* {{cite journal |url=http://circ.ahajournals.org/cgi/reprint/102/18/2214 |author=Francis|title=Quantitative General Theory for Periodic Breathing |journal=Circulation |volume=102 |issue=18 |pages=2214–21 |year=2000 |doi=10.1161/01.cir.102.18.2214 |pmid=11056095|display-authors=etal|citeseerx=10.1.1.505.7194}} (Lambert function is used to solve delay-differential dynamics in human disease.)
* {{Cite journal | last=Hayes | first=B. | title=Why ''W''? | url=https://www.americanscientist.org/sites/americanscientist.org/files/2005216151419_306.pdf | year=2005 | journal=American Scientist | volume=93 | issue=2 | pages=104–108 | doi=10.1511/2005.2.104}}
* {{dlmf|id=4.13|title=Lambert W function|first=R. |last=Roy|first2=F. W. J. |last2=Olver}}
* {{cite journal |last1=Stewart |first1=Seán M. |year=2005 |title=A New Elementary Function for Our Curricula? |url=http://files.eric.ed.gov/fulltext/EJ720055.pdf |format=PDF |journal=Australian Senior Mathematics Journal |volume=19 |issue=2 |pages=8–26 |issn=0819-4564 |id=[[Education Resources Information Center|ERIC]] EJ720055 |layurl=http://eric.ed.gov/?id=EJ720055 }}
* [https://arxiv.org/abs/1003.1628 Veberic, D., "Having Fun with Lambert W(x) Function" arXiv:1003.1628 (2010)]; {{Cite journal | last1=Veberic | first1=D. | title=Lambert W function for applications in physics | year=2012 | journal=Computer Physics Communications | volume=183 | issue=12 | pages=2622–2628 | doi=10.1016/j.cpc.2012.07.008 | arxiv=1209.0735 | bibcode=2012CoPhC.183.2622V }}
* {{cite journal | arxiv=1601.04895| last1 = Chatzigeorgiou | first1 = I. |title=Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation |journal=IEEE Communications Letters |volume=17 |issue=8 |pages=1505–1508 |year=2013 |doi=10.1109/LCOMM.2013.070113.130972}}

==External links==
{{commons category}}
* [http://dlmf.nist.gov/4.13 National Institute of Science and Technology Digital Library – Lambert W]
* [http://mathworld.wolfram.com/LambertW-Function.html MathWorld – Lambert ''W''-Function]
* [https://web.archive.org/web/20060824060948/http://www.whim.org/nebula/math/lambertw.html Computing the Lambert W function]
* [http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/ Corless et al. Notes about Lambert W research]
* GPL [https://github.com/DarkoVeberic/LambertW C++ implementation] with Halley's and Fritsch's iteration.
* [https://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html Special Functions] of the [https://www.gnu.org/software/gsl/ GNU Scientific Library] – GSL

{{DEFAULTSORT:Lambert W Function}}
[[Category:Special functions]]