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| [[Image:Lambert-w.svg|thumb|288px|right|The graph of ''W''(''x'') for ''W'' > −4 and ''z'' < 6. The upper branch with ''W'' ≥ −1 is the function ''W''<sub>0</sub> (principal branch), the lower branch with ''W'' ≤ −1 is the function ''W''<sub>−1</sub>.]]
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| In [[mathematics]], the '''Lambert W function''', also called the '''omega function''' or '''product logarithm''', is a set of [[function (mathematics)|functions]],
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| namely the branches of the [[inverse relation]] of the function ''z'' = ''f''(''W'') = ''We''<sup>''W''</sup> where ''e''<sup>''W''</sup> is the [[exponential function]] and ''W'' is any [[complex number]]. In other words, the defining equation for ''W''(''z'') is
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| :<math>z = W(z)e^{W(z)}</math>
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| for any complex number ''z''.
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| Since the function ''ƒ''<math>\scriptstyle (\cdot)</math> is not [[injective]], the relation ''W'' is [[multivalued function|multivalued]] (except at 0). If we restrict attention to real-valued ''W'', the complex variable ''z'' is then replaced by the real variable ''x'', and the relation is defined only for ''x'' ≥ −1/''e'', and is double-valued on (−1/''e'', 0). The additional constraint ''W'' ≥ −1 defines a single-valued function ''W''<sub>0</sub>(''x''). We have ''W''<sub>0</sub>(0) = 0 and ''W''<sub>0</sub>(−1/''e'') = −1. Meanwhile, the lower branch has ''W'' ≤ −1 and is denoted ''W''<sub>−1</sub>(''x''). It decreases from ''W''<sub>−1</sub>(−1/''e'') = −1 to ''W''<sub>−1</sub>(0<sup>−</sup>) = −∞.
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| The Lambert ''W'' relation cannot be expressed in terms of [[elementary function]]s.<ref>{{citation
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| | last = Chow | first = Timothy Y.
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| | doi = 10.2307/2589148
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| | issue = 5
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| | journal = [[American Mathematical Monthly]]
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| | mr = 1699262
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| | pages = 440–448
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| | title = What is a closed-form number?
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| | volume = 106
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| | year = 1999}}.</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 ''y'''(''t'') = ''a'' ''y''(''t'' − 1).
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| :[[Image:Product Log.jpg|thumb|288px|right|Main branch of the Lambert ''W'' function in the complex plane. Note the [[branch cut]] along the negative real axis, ending at −1/''e''. In this picture, the hue of a point ''z'' is determined by the [[argument (complex analysis)|argument]] of ''W''(''z'') and the brightness by the [[absolute value]] of ''W''(''z'').]]
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| ==Terminology==
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| [[File:Diagram of the real branches of the Lambert W function.png|thumb|The two main branches <math>W_0</math> and <math>W_{-1}</math>]]
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| The Lambert ''W''-function is named after [[Johann Heinrich Lambert]]. The main branch ''W''<sub>0</sub> is denoted by ''Wp'' in the [[Digital Library of Mathematical Functions]] and the branch ''W''<sub>−1</sub> is denoted by ''Wm'' there.
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| The notation convention chosen here (with ''W''<sub>0</sub> and ''W''<sub>−1</sub>) follows the canonical reference on the Lambert-''W'' function by Corless, Gonnet, Hare, Jeffrey and [[Donald Knuth|Knuth]].<ref name = "Corless">{{cite journal
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| | last1 = Corless | first1 = R. M.
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| | last2 = Gonnet | first2 = G. H.
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| | last3 = Hare | first3 = D. E. G.
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| | last4 = Jeffrey | first4 = D. J.
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| | last5 = Knuth | first5 = D. E.
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| | author5-link = Donald Knuth
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| | title = On the Lambert W function
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| | journal = Advances in Computational Mathematics
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| | volume = 5
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| | pages = 329–359
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| | year = 1996
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| | url = http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps
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| | doi = 10.1007/BF02124750
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| }}</ref>
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| ==History==
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| Lambert first considered the related ''Lambert's Transcendental Equation'' in 1758,<ref>Lambert JH, "Observationes variae in mathesin puram", ''Acta Helveticae physico-mathematico-anatomico-botanico-medica'', Band III, 128–168, 1758 ([http://www.kuttaka.org/~JHL/L1758c.pdf facsimile])</ref> which led to a paper by [[Leonhard Euler]] in 1783<ref>Euler, L. "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. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref> that discussed the special case of ''we<sup>w</sup>''. However the inverse of ''we<sup>w</sup>'' was first described by Pólya and Szegő in 1925.<ref name="Pólya">{{cite book
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| | last1 = Pólya | first1 = George
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| | last2 = Szegő | first2 = Gábor
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| | author1-link = George Pólya
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| | author2-link = Gábor Szegő
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| | origyear = 1925
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| | title = Aufgaben und Lehrsätze der Analysis
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | trans_title = Problems and Theorems in Analysis |year=1998
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| }}</ref>{{Citation needed|reason=Volume and page number missing in Pólya/Szegő|date=October 2012}} The Lambert W function was "re-discovered" every decade or so in specialized applications but its full importance was not realized until the 1990s. When it was reported that the Lambert 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 to find that this function was in fact ubiquitous to 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 |publisher=MapleTech |volume=9 |issue= |pages=12–22 |year=1993 |url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.2556&rep=rep1&type=pdf}}</ref>
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| ==Calculus==
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| ===Derivative===
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| By [[implicit differentiation]], one can show that all branches of ''W'' satisfy the [[ordinary differential equation|differential equation]]
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| :<math>z(1+W)\frac{{\rm d}W}{{\rm d}z}=W\quad\text{for }z\neq -1/e.</math>
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| (''W'' is not differentiable for ''z'' = −1/''e''.) As a consequence, we get the following formula for the derivative of ''W'':
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| :<math>\frac{{\rm d}W}{{\rm d}z}=\frac{W(z)}{z(1 + W(z))}\quad\text{for }z\not\in\{0,-1/e\}.</math>
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| Furthermore we have
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| :<math>\left.\frac{{\rm d}W}{{\rm d}z}\right|_{z=0}=1.</math>
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| ===Antiderivative===
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| The function ''W''(''x''), and many expressions involving ''W''(''x''), can be [[integral|integrated]] using the [[substitution rule|substitution]] ''w'' = ''W''(''x''), i.e. ''x'' = ''w'' e<sup>''w''</sup>:
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| :<math>\int W(x)\,{\rm d}x = x \left( W(x) - 1 + \frac{1}{W(x)} \right) + C.</math>
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| ==Asymptotic expansions==
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| The [[Taylor series]] of <math>W_0</math> around 0 can be found using the [[Lagrange inversion theorem]] and is given by
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| :<math>
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| W_0 (x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}\ x^n = x - x^2 + \frac{3}{2}x^3 - \frac{8}{3}x^4 + \frac{125}{24}x^5 - \cdots
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| </math>
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| The [[radius of convergence]] is 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]] (−∞, −1/''e''<nowiki>]</nowiki>; this holomorphic function defines the [[principal branch]] of the Lambert ''W'' function.
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| For large values of ''x'', ''W''<sub>0</sub> is asymptotic to
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| :<math>
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| W_{0} (x) = L_1 - L_2 + \frac{L_2}{L_1} + \frac{L_2 (-2 + L_2)}{2 L_1^2} + \frac{ L_2 (6 - 9 L_2 + 2 L_2^2) }{6 L_1^3} + \frac{L_2 (-12+36L_2 - 22 L_2^2 + 3 L_2^3)}{12 L_1^4} + \cdots
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| </math>
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| :<math>
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| W_{0} (x) = L_1-L_2+\sum_{\ell=0}^{\infty}\sum_{m=1}^{\infty}\frac{(-1)^{\ell}\left [\begin{matrix} \ell+m \\ \ell + 1 \end{matrix}\right ]}{m!} L_1^{-\ell-m} L_2^{m}
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| </math>
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| where <math>L_1=\ln x</math>, <math>L_2=\ln\ln x</math> and <math>\left [\begin{matrix} \ell+m \\ \ell + 1 \end{matrix}\right ]</math> is a non-negative [[Stirling numbers of the first kind|Stirling number of the first kind]].<ref>[http://rgmia.org/papers/v10n2/lambert-v2.pdf Approximation of the Lambert W function and the hyperpower function], Hoorfar, Abdolhossein; Hassani, Mehdi.</ref> Keeping only the first two terms of the expansion,
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| :<math>W_0(x)=\ln x-\ln\ln x+o(1).</math>
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| The other real branch, <math>W_{-1}</math>, defined in the interval (−∞, −1/''e''<nowiki>]</nowiki>, has an approximation of the same form as ''x'' approaches zero, with in this case
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| <math>L_1=\ln(-x)</math> and <math>L_2=\ln(-\ln(-x))</math>.
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| ===Integer and complex powers===
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| Integer powers of <math>W_0</math> also admit simple Taylor (or [[Laurent series|Laurent]]) series expansions at <math>0</math>
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| :<math>
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| W_0(x)^2 = \sum_{n=2}^\infty \frac{-2(-n)^{n-3}}{(n-2)!}\ x^n = x^2-2x^3+4x^4-\frac{25}{3}x^5+18x^6- \cdots
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| </math>
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| More generally, for <math>r\in\Z,</math>, the [[Formal power series#The Lagrange inversion formula|Lagrange inversion formula]] gives
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| :<math>
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| W_0(x)^r = \sum_{n=r}^\infty \frac{-r(-n)^{n-r-1}}{(n-r)!}\ x^n,
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| </math>
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| which is, in general, a Laurent series of order ''r''. Equivalently, the latter can be written in the form of a Taylor expansion of powers of <math>W_0(x)/x</math>
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| :<math>
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| \left(\frac{W_0(x)}{x}\right)^r =\exp(-r W_0(x)) = \sum_{n=0}^\infty \frac{r(n+r)^{n-1}}{n!}\ (-x)^n,
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| </math>
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| which holds for any <math>r\in\C</math> and <math>|x|<e^{-1}</math>.
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| ==Special values==
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| For any non-zero [[algebraic number]] ''x'', ''W''(''x'') is a [[transcendental number]]. We can show this by [[Proof by contradiction|contradiction]]: If ''W''(''x'') were non-zero and algebraic (note that if ''x'' is non-zero then ''W''(''x'') must be non-zero as well), then by the [[Lindemann–Weierstrass theorem]], ''[[e (mathematical constant)|e]]''<sup>''W''(''x'')</sup> must be transcendental, implying that ''x''=''W''(''x'')''e''<sup>''W''(''x'')</sup> must also be transcendental, contradicting the condition that ''x'' is algebraic.
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| <math>W\left(-\frac{\pi}{2}\right) = \frac{\pi}{2}{\rm{i}} </math>
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| <math>W\left(-\frac{\ln a}{a}\right)= -\ln a \quad \left(\frac{1}{e}\le a\le e\right)</math>
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| <math>W\left(-\frac{1}{e}\right) = -1</math>
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| <math>W\left(0\right) = 0\,</math>
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| <math>W\left(1\right) = \Omega=\frac{1}{\displaystyle \int_{-\infty}^{+\infty}\frac{\,dt}{(e^t-t)^2+\pi^2}}-1\approx 0.56714329\dots\,</math> (the [[Omega constant]])
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| <math>W\left(e\right) = 1\,</math>
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| <math>W\left(-1\right) \approx -0.31813-1.33723{\rm{i}} \,</math>
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| <math>W'\left(0\right) = 1\,</math>
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| == Other formulas ==
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| :<math>\int_{0}^{\pi} W\bigl( 2\cot^2(x) \bigr)\sec^2(x)\;\mathrm dx = 4\sqrt{\pi}</math>
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| :<math>\int_{0}^{+\infty} W\left(\frac{1}{x^2}\right)\;\mathrm dx = \sqrt{2\pi}</math>
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| :<math>
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| \int_{0}^{+\infty} \frac{W(x)}{x\sqrt{x}}\mathrm dx = 2\sqrt{2\pi}
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| </math>
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| ==Applications==
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| Many [[equation]]s involving exponentials can be solved using the ''W'' function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like ''Y'' = ''Xe''<sup>''X''</sup> at which point the ''W'' function provides the value of the variable in ''X''.
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| In other words :
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| :<math> Y = X e ^ X \; \Longleftrightarrow \; X = W(Y) </math>
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| ===Examples===
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| ==== Example 1 ====
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| <math>
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| \begin{align}
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| 2^t &= 5t\\
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| 1 &= \frac{5 t}{2^t}\\
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| 1 &= 5 t \, e^{-t \ln 2}\\
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| \frac{1}{5} &= t \, e^{-t \ln 2}\\
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| \frac{- \, \ln 2}{5} &= ( - \, t \, \ln 2 ) \, e^{( -t \ln 2 )}\\
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| W \left ( \frac{- \ln 2}{5} \right ) &= -t \ln 2\\
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| t &= -\frac{W \left ( \frac{- \ln 2}{5} \right )}{\ln 2}
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| \end{align}
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| </math>
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| More generally, the equation
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| : <math> ~p^{a x + b} = c x + d </math>
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| where
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| : <math> p > 0 \text{ and } c,a \neq 0 </math>
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| can be transformed via the substitution
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| : <math> -t = a x + \frac{a d}{c} </math>
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| into
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| : <math> t p^t = R = -\frac{a}{c} p^{b-\frac{a d}{c}} </math>
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| giving
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| : <math> t = \frac{W(R\ln p)}{\ln p} </math>
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| which yields the final solution
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| : <math> x = -\frac{W(-\frac{a\ln p}{c}\,p^{b-\frac{a d}{c}})}{a\ln p} - \frac{d}{c} </math>
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| ==== Example 2 ====
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| :<math>x^x=z\,</math>
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| :<math>\Rightarrow x\ln x = \ln z\,</math>
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| :<math>\Rightarrow e^{\ln x} \cdot \ln x = \ln z\,</math>
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| :<math>\Rightarrow \ln x = W(\ln z)\,</math>
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| :<math>\Rightarrow x=e^{W(\ln z)}\, ,</math>
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| or, equivalently,
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| :<math>x=\frac{\ln z}{W(\ln z)},</math>
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| since
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| : <math>\ln z = W(\ln z) e^{W(\ln z)}\,</math>
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| by definition.
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| ==== Example 3 ====
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| Whenever the complex infinite exponential [[tetration]]
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| :<math>z^{z^{z^{\cdot^{\cdot^{\cdot}}}}} \!</math>
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| converges, the Lambert ''W'' function provides the actual limit value as
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| :<math>c=\frac{W(-\ln(z))}{-\ln(z)}</math>
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| where ln(''z'') denotes the principal branch of the complex log function.
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| ==== Example 4 ====
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| Solutions for
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| :<math>x \log_b \left(x\right) = a</math>
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| have the form <ref name="corless_maple"/>
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| :<math>x=e^{W(a\ln b)}.</math>
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| ==== Example 5 ====
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| The solution for the [[Electric current|current]] in a series [[diode]]/[[resistor]] [[Electrical network|circuit]] can also be written in terms of the Lambert W. See [[Diode modelling#Explicit solution|diode modeling]].
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| ==== Example 6 ====
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| The [[delay differential equation]]
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| :<math>\dot{y}(t) = ay(t-1)</math>
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| has [[Delay differential equation#The characteristic equation|characteristic equation]] <math>\lambda=a e^{-\lambda}</math>, leading to <math>\lambda=W_k(a)</math> and <math>y(t)=e^{W_k(a)t}</math>, where <math>k</math> is the branch index. If <math>a \ge e^{-1}</math>, only <math>W_0(a)</math> need be considered.
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| ==== Example 7 ====
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| The Lambert ''W'' function has been recently shown to be the optimal solution for the required magnetic field of a [[Zeeman slower]].<ref>{{cite journal |last1= B Ohayon. |first1=G Ron. |title=New approaches in designing a Zeeman Slower |journal=Journal of Instrumentation |volume=8 |issue=02 |pages=P02016 |year=2013 |doi=10.1088/1748-0221/8/02/P02016 }}</ref>
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| ==== Example 8 ====
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| Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in the laboratory experiments can be described by using the Lambert–Euler omega function as follows:
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| :<math>H(x)= 1 + W[(H(0) -1) \exp((H(0)-1)-\frac{x}{L})],</math>
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| where ''H(x)'' is the debris flow height, ''x'' is the channel downstream position, ''L'' is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.
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| ==== Example 9 ====
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| The Lambert ''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}}</ref>
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| ==Generalizations==
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| The standard Lambert ''W'' function expresses exact solutions to ''transcendental algebraic'' equations (in ''x'') of the form:
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| :<math> e^{-c x} = a_o (x-r) ~~\quad\qquad\qquad\qquad\qquad(1)</math>
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| where ''a''<sub>0</sub>, ''c'' and ''r'' are real constants. The solution is <math> x = r + \frac{1}{c} W\!\left( \frac{c\,e^{-c r}}{a_o } \right)\, </math>. Generalizations of the Lambert ''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 }}</ref> include:
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| *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 <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 }}</ref>) between these two areas, where the right-hand-side of (1) is now a quadratic polynomial in ''x'':
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| :<math> e^{-c x} = a_o (x-r_1 ) (x-r_2 ) ~~\qquad\qquad(2)</math>
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|
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| :and where ''r''<sub>1</sub> and ''r''<sub>2</sub> are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function has a single argument ''x'' but the terms like ''r''<sub>i</sub> and ''a''<sub>o</sub> are parameters of that function. In this respect, the generalization resembles the [[hypergeometric]] function and the [[Meijer G-function]] but it belongs to a different ''class'' of functions. When ''r''<sub>1</sub> = ''r''<sub>2</sub>, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard ''W'' function. Eq. (2) expresses the equation governing the [[dilaton]] field, from which is derived the metric of the [[R=T model|R=T]] or ''lineal'' two-body gravity problem in 1+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.
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| *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 }}</ref> Here the right-hand-side of (1) (or (2)) is now a ratio of infinite order polynomials in ''x'':
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| ::<math> e^{-c x} = a_o \frac{\displaystyle \prod_{i=1}^{\infty} (x-r_i )}{\displaystyle \prod_{i=1}^{\infty} (x-s_i)} \qquad \qquad\qquad(3)</math>
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| :where ''r''<sub>i</sub> and ''s''<sub>i</sub> are distinct real constants and ''x'' is a function of the eigenenergy and the internuclear distance ''R''. Eq. (3) with its specialized cases expressed in (1) and (2) is related to a large class of [[delay differential equation]]s.
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| Applications of the Lambert "W" function in fundamental physical problems are not exhausted even for the standard case expressed in (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 }}</ref>
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| ==Plots==
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| <gallery caption="Plots of the Lambert W function on the complex plane">
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| Image:LambertWRe.png| ''z'' = Re(W<sub>0</sub>(''x'' + ''i'' ''y''))
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| Image:LambertWIm.png| ''z'' = Im(W<sub>0</sub>(''x'' + ''i'' ''y''))
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| Image:LambertWAbs.png| ''z'' = |W<sub>0</sub>(''x'' + ''i'' ''y'')|
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| Image:LambertWAll.png
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| </gallery>
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| ==Numerical evaluation==
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| The ''W'' function may be approximated using [[Newton's method]],
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| with successive approximations to <math>w=W(z)</math> (so <math>z=we^w</math>) being | |
| :<math>w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}+w_j e^{w_j}}.</math>
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| The ''W'' function may also be approximated using [[Halley's method]],
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| :<math>
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| w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}(w_j+1)-\frac{(w_j+2)(w_je^{w_j}-z)}
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| {2w_j+2}}
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| </math>
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| given in Corless et al. to compute ''W''.
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| == Software ==
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| The LambertW function is implemented as
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| [http://www.maplesoft.com/support/help/Maple/view.aspx?path=LambertW LambertW in Maple], <code>lambertw</code> in [[PARI/GP|GP]] (and <code>glambertW</code> in PARI), <code>lambertw</code> in MATLAB,<ref>[http://www.mathworks.com.au/help/toolbox/symbolic/lambertw.html lambertw - MATLAB]</ref> also <code>lambertw</code> in octave with the 'specfun' package, as <code>lambert_w</code> in Maxima<ref>[http://maxima.sourceforge.net Maxima, a Computer Algebra System]</ref>, as <code>ProductLog</code> (with a silent alias <code>LambertW</code>) in Mathematica,<ref>[http://reference.wolfram.com/mathematica/ref/ProductLog.html ProductLog at WolframAlpha]</ref> and as <code>gsl_sf_lambert_W0</code> and <code>gsl_sf_lambert_Wm1</code> functions in [http://www.gnu.org/software/gsl/manual/html_node/Lambert-W-Functions.html special functions] section of the [http://www.gnu.org/software/gsl/ GNU Scientific Library] - GSL. | |
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| ==See also==
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| * [[Wright Omega function]]
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| * Lambert's [[trinomial|trinomial equation]]
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| * [[Lagrange_inversion_theorem#Lambert_W_function|Lagrange inversion theorem]]
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| * [[Experimental mathematics]]
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| * [[Holstein-Herring method]]
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| * [[R=T model]]
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| * [[Ross' π lemma]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * {{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 | publisher=[[Springer-Verlag]] | location=Berlin, New York | 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}}}}
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| * {{cite journal |url=http://www.istia.univ-angers.fr/~chapeau/papers/lambertw.pdf |author=Chapeau-Blondeau, F. and Monir, A. |title=Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2 |journal=IEEE Trans. Signal Processing |volume=50 |issue=9 |year=2002}}
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| * {{cite journal |url=http://circ.ahajournals.org/cgi/reprint/102/18/2214 |author=Francis et al. |title=Quantitative General Theory for Periodic Breathing |journal=Circulation |volume=102 |issue=18 |pages=2214 |year=2000}} (Lambert function is used to solve delay-differential dynamics in human disease.)
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| * {{Cite journal | last=Hayes | first=B. | title=Why ''W''? | url=http://www.americanscientist.org/issues/pub/why-w | year=2005 | journal=American Scientist | volume=93 | pages=104–108}}
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| * {{dlmf|id=4.13|title=Lambert W function|first=R. |last=Roy|first2=F. W. J. |last2=Olver}}
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| * [http://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 | pages=2622–2628 | doi=10.1016/j.cpc.2012.07.008 | postscript=. }}
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| ==External links==
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| * [http://dlmf.nist.gov/4.13 National Institute of Science and Technology Digital Library - Lambert W]
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| * [http://mathworld.wolfram.com/LambertW-Function.html MathWorld - Lambert ''W''-Function]
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| * [http://www.whim.org/nebula/math/lambertw.html Computing the Lambert W function]
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| * [http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/ Corless et al. Notes about Lambert W research]
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| * [http://ioannis.virtualcomposer2000.com/math/ Extreme Mathematics.] Monographs on the Lambert ''W'' function, its numerical approximation and generalizations for ''W''-like inverses of transcendental forms with repeated exponential towers.
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| * GPL [http://www.ung.si/~darko/LambertW/ C++ implementation] with Halley's and Fritsch's iteration.
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| * [http://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html Special Functions] of the [http://www.gnu.org/software/gsl/ GNU Scientific Library] - GSL
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| {{DEFAULTSORT:Lambert W Function}}
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| [[Category:Special functions]]
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