Talk:Lambert W function

message: Details to solve x^x = z.
The trick is to express x as e^{\ln(x)}.
x^x = z  
(e^{\ln(x)})^{(e^{\ln(x)})} = z  
e^{\ln(x) e^{\ln(x)}} = z  
\ln(x) e ^ {\ln(x)} = \ln(z)  
\ln(x) = W(\ln(z))  
x = e ^ {W(\ln(z))}  
x = \ln(z)/W(\ln(z))  
Corwinjoy (talk) 19:48, 6 August 2010 (UTC)
replies: text: Is it similarly possible to solve   x^{x+1} = 2   ?   (x = 1.344070114603687...)   Is there an explicit expression for x? -- MathLine (talk) 15:57, 11 June 2013 (UTC)
thread_title:  Example 2 

message: Can we get an image of a fractal related to the Lambert W fun?  If anyone has one (or can construct one), this article would benefit from its inclusion.  (I know that fractals are not why Lambert W is important, and that Lambert W has more benefit as an equation implicitly defined by an elementary equation though it, itself is not an elementary equation, however a picture (of a fractal), and its ascetic beauty cultivate further interest in this most interesting function).
-- LinuxDude 17:20, 8 January 2007 (UTC)
thread_title:  Request 

message: Can someone include input() and print statements to get output from that Python code?  —Preceding unsigned comment added by Charlesrkiss (talk • contribs) 17:58, 5 April 2008 (UTC)
thread_title:  Python Algorithm 

message: (arxiv articles and journal papers):
[AAECC]:  http://arxiv.org/abs/math-ph/0607011
          http://portal.acm.org/citation.cfm?id=1127202.1127208&coll=&dl=ACM
[gravity/QM]:  http://arxiv.org/abs/gr-qc/0611144
no. of [4647] in http://www.iop.org/EJ/toc/0264-9381/24/18
[H2+]:    http://arxiv.org/abs/physics/0607081
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TFM-4HNYMS6-5&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=9fd01e7be3137ccf30280c1281b62e14
OK, if its INCOMPLETE and NEEDS TO BE UPDATED, why don't you be bold and add the requisite material yourself?  I have Scott, Mann, and Martinez right here, so go ahead!  Best wishes, Robinh (talk) 19:40, 5 December 2008 (UTC)
thread_title: References

message: The new material looks good.  But I am wondering (as I was after reading Scott et al) what exactly the generalization actually is.  Is it a function of three arguments (x, r1, r2)?  The standard function has W(x)*exp(W(x)) = x.  Is there a similar defining identity for the generalized version?  Yours in confusion, Robinh (talk) 08:00, 11 December 2008 (UTC)
thread_title: Generalization

message: Am not sure if this is the right place to make queries about productlog usage but perhaps someone can help?!
W(5.e^5)=5
Is there a simplification for:
W(-5.e^-5)
Where W denotes the product log function?
Neil Parker (talk) 19:15, 20 May 2009 (UTC)
replies: text: See the equation  Y = X e ^ X \; \Longleftrightarrow \; X = W(Y)  in the article: Y=-5e^{-5} means W(Y)=-5. Shreevatsa (talk) 22:04, 20 May 2009 (UTC)

text: Please consult the reference desk for questions like this. — Tobias Bergemann (talk) 18:43, 23 May 2009 (UTC)
thread_title:  Need help simplifying 

message: Is d=/=0 really required here? I do not see where that requirement comes from.  —Preceding unsigned comment added by 77.4.43.178 (talk) 23:49, 22 June 2009 (UTC)
thread_title:  p^(ax+b)=cx+d 

message: From the article:
:However the inverse of wew was first described by Pólya and Szegö in 1925.
This could only refer to "Problems and Theorems in Analysis", originally published as
 G Pólya and G Szegö, Aufgaben und Lehrsätze aus der Analysis, Springer 1925.
But I haven't read it so I'm not going to add this as a reference.
CRGreathouse (t | c) 19:18, 17 August 2009 (UTC)
thread_title:  Reference 

message: Now I understand why they did not yet find supersymmetry at DESY - desy_lambert_W() is an extremely rough approximation to LambertW() !
In x=1 (where W(1) = 0.5...), they differ by -0.06... this may seem little for everyday use, but for a scientific purpose, that's not serious ! In addition, it really does not make sense to declare it as "double precision". — MFH:Talk 16:36, 13 November 2009 (UTC)
thread_title:  DESY Lambert W 

message: I suggest reverting , although the new notation could be noted as well. I have only seen this notation used in the DLMF, and while the DLMF can be considered authoritative, I think the notation of Corless et al. should be used until/if the DLMF notation becomes more widespread. One problem with p/m or +/- is that it is limited to only two branches, but the numerical subscript allows indexing all branches of the Lambert W function (this article fails to note this). The "old" notation is also used in computer algebra systems, which are the most important domain of use for the Lambert W function.
In fact, a part of the article says "characteristic equation \lambda=a e^{-\lambda}, leading to \lambda=W_k(a) and y(t)=e^{W_k(a)t}, where k is the branch index.  If a is real, only W_0(a) need be considered" which is unintelligible right now when W_0 (and the general branches) aren't defined. Fredrik Johansson 04:39, 6 June 2010 (UTC)
thread_title:  DLMF notation 

message: Is it just me, or does anyone else realize that the Lambert W function is not a function?!  —Preceding unsigned comment added by 74.111.112.90 (talk) 19:31, 29 August 2010 (UTC)
replies: text: It is common to call something a function when its branch cuts are functions. There is no confusion when it is used this way. Most people know for instance that the square root is a relation, but they know also that by convention  \sqrt{x}  usually only refers to the principal branch and it is also generally called a function.--75.80.43.80 (talk) 14:51, 3 March 2011 (UTC)
thread_title: Moving to Lambert W relation=

message: At  it says that W(2)i (4/π)i is the Definite integral over a half-period of i^x. Perhaps this shold be in this artical? Robo37 (talk) 22:30, 18 August 2011 (UTC)
thread_title:  Why is this? 

message: I took out this line:
:exp(-W\left(0\right))(-ln(3^{1/3}) = 2.47805268028830\dots, the solution of y of  \sqrt[x]{x}= \sqrt[y]{y} when x=3 and x≠y.
because (1) the math is badly formatted and it's unclear what is meant: the parentheses don't match up. (2) W(0)=0, so why this is in the expression is unclear. (3) What's so special about x=3 or \sqrt[x]{x}= \sqrt[y]{y} that this should be listed in this article? If you disagree with the removal, please discuss here. Thanks!—24.85.177.92 (talk) 14:57, 23 August 2011 (UTC)
replies: text: Well the number is listed at oeis and other websites, and the 3 just arises because we have 4 as the solution to x=2 and 2 as the solution to x=4, so 3 just kind of sits nicely in the middle, though I'm not sure whether that makes it notable enough or not. Maybe -2W({1 \over 2}) which is the solution to ex=x2, -iW(i) which is the solution to ex=xi, \ln{2} \over 2 which is the solution to W(x)=2x and/or  \sqrt {W(2) \over 2} which is the solution to W(x)=x2 should be included as well/instead, as there representations are simpler? Robo37 (talk) 19:18, 26 August 2011 (UTC)
thread_title:  Deleted a line from special values 

message: The statement "Corless and developers of the Maple Computer algebra system made a library search to find that this function was in fact ubiquitous to nature" isn't clear because "library" could refer to a conventional library of printed books or it could refer to software libraries.  I don't know how a search of either could prove something about Nature.  I think the statement should be clarified or omitted.
Tashiro (talk) 14:59, 23 October 2011 (UTC)
replies: text: I find nothing wrong with the statement.  Given that dates of their papers (early 1990s) proficiency with the web was not that well established back then.   Wikipedia came later.  Clearly these were bonafide "library" searches as in e.g. University library searches.  Their initial article with appeared in MapleTech showed that the Lambert W function appeared in a number of applications demonstrating the universality of the Lambert W function.  What is not clear about this? TonyMath (talk) 18:00, 14 September 2012 (UTC)
replies: text: I think I understand in part the nature of your objection.  It's because the notion that the work of Corless et al being apparently "reduced" to a library search seems inglorious and lacking in respect.  A couple of points should be made: (i) that search was absolutely essential in demonstrating that the W function was fundamental in nature - we needed to know that - and (ii) their results were really not so original after all.  In particular long before they had examined the W function in the complex plane, mathematical physicists like Byers-Brown  had already sussed that out in the context of the  Double well Dirac Delta function model in itself a 1-D representation of the Hydrogen molecular ion for a generalized case back in the 1970s!  It stands to reason: the Lambert W function is centuries old and thus it becomes virtually impossible for any of us to have a full assessment of all the developments involving that function.  We must remain objective when examining things with hindsight. TonyMath (talk) 23:29, 30 September 2012 (UTC)

text: I agree with the above comment regarding "ubiquitous in nature". The paper cited reads "it has appeared in the solution of enough problems to have earned the right to have a name of its own". A more precise citation is needed for the statement in the article. Isheden (talk) 12:16, 1 October 2012 (UTC)
replies: text: Done.  Corless' main paper does indeed talk about the many many applications of the Lambert W function though his paper cites only a few.TonyMath (talk) 15:55, 2 October 2012 (UTC)
replies: text: I should also add that the wikipedia article itself with some 8 examples + generalizations also makes the case of the wide range of applications of the W function.  Even Corless' paper cannot do full justice to that notion.  I think the W function could use a counterpart of the Fibonacci quaterly but for Lambert's function TonyMath (talk) 15:59, 2 October 2012 (UTC)
thread_title: Meaning of "a library search"?

message: In the section special values, it says
W\left(-\frac{\ln a}{a}\right)= -\ln a  \quad            \left(\frac{1}{e}\le a\le e\right)
I believe this relation holds true for 0<a\le e, can anyone confirm this?
Tjajab (talk) 12:22, 30 December 2012 (UTC)
thread_title:  Question about the range of one of the special values 

message: -W(-1) is an fixed point of the exponential function. Here's the proof:
:-1=W(-1) e^{W(-1)}
:-W(-1)^{-1}= e^{W(-1)}
:-W(-1) = e^{-W(-1)}
So z=-W(-1) is z which satisfies e^z=z, and so is its complex conjugate. --31.147.15.123 (talk) 11:32, 8 October 2013 (UTC)
thread_title:  Fixed point of the exponential function 

message: Product rule
ω(a)*ω(b)*exp[ω(a)+ω(b)]=a*b=ω(a*b)*exp[ω(a+b)]
Solve for ω(a*b)
ω(a)*ω(b)=c*exp(c)
exp[ω(a)+ω(b)]=d*exp(d)
c=ω[ω(a)*ω(b)]
d=ω[exp[ω(a)+ω(b)]]
ω(a*b)=c*d=ω[ω(a)*ω(b)]*ω[exp[ω(a)+ω(b)]]
For instance:
ω(2*3)=ω[ω(2)*ω(3)]*ω[exp[ω(2)+ω(3)]]
4.49.117.146 (talk) 22:14, 13 February 2014 (UTC)
thread_title:  Product rule for Omega function 

message: Second term (n=2), gives denominator of 2 via the summation which is not what is in the expansion. Cross-checking with Wolfram Mathworld shows a difference. --Paddy (talk) 07:28, 30 August 2014 (UTC)
thread_title:  W0 Taylor series expansion is wrong 

message: Hi, I want to solve the following equation: (x^2-4*x+6)*exp(x)=y (1) It looks a bit like the following equation: x*exp(x)=y (2) The solution of equation (2) is: x=LambertW(y) I think the solution of equation (1) uses the function LambertW. Can somebody solve the equation (1)? Kindest regards, Jacob Safra  — Preceding unsigned comment added by Jacobsafra (talk • contribs) 01:06, 7 October 2014 (UTC)
replies: text: (1) is simply exp(-x) = (1/y)(x^2-4*x+6).  This is a special case of eq.(2) of the generalized Lambert W function, the right-hand-side being a quadratic polynomial but with complex roots i.e. 2 +- sqrt(-2).   TonyMath (talk) 01:27, 12 October 2014 (UTC)
replies: text: In general there will be no closed form solution. After all, this is a transcendental-algebraic equation. Using methods  currently developed by a colleague, I can find some exact solutions.  E.g. x=1-i, y=(2+2*i)*e^(1-i) and x=1+i, y=(2-2*i)*e^(1+i).  These solutions are in the complex plane however. There may be other exact solutions. Does this originate from a physics or engineering problem? TonyMath (talk) 11:49, 26 October 2014 (UTC)

text: Getting interesting.  Unless I misunderstand, your 'g' is just (or is related to) the inverse generalized Lambert W function and 'x' is just the solution.  I think Mathematica has an inverse for the standard Lambert W function but nobody, as far as I know, has defined an inverse for the generalized Lambert W function.  The bottom line, I think, is that you are looking for exact solutions.  I passed on your problem to that colleague.  Maybe she can find something.  I would like to know more about the physical problem.  Is there a reference for it? or is this a physical model you have been developing on your own?  At any rate, the SIGSAM papers in the references could provide (i) series solutions to this problem and (ii) a numerical method for getting ALL the roots.  Unfortunately they are not yet implemented on a general computer algebra system. TonyMath (talk) 03:10, 28 October 2014 (UTC)
replies: text: So far, my colleague managed to prove that the solution to your problem is unique, but no closed form solution so far. TonyMath (talk) 03:02, 14 November 2014 (UTC)
thread_title:  Request for help to solve an equation 

message: The discussion on branches and what they mean seems a bit sparse and confusing (there are 2 or 3 lines in the summary). From other sources, it appears there exists a number of branches W_k. A more involved discussion on W_o and W_-1 and W_k would be great. If some one could add a section that discusses the branches, it would improve the article. I don't have the expertise to do so. Gsonnenf (talk) 02:22, 30 September 2015 (UTC)
thread_title:  Discussion of branches 

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thread_title:  External links modified 

message: Should we add a formula for the -1 branch of the Lambert W Function? I only see formulas for the 0 branch. 63.227.221.214 (talk) 20:14, 5 May 2021 (UTC)
thread_title:  -1 branch of W function 

message: I think it would be a good idea to create a table in the Software section of this article. It could make it easier to read. Somerandomuser (talk) 05:14, 17 September 2021 (UTC)
thread_title:  Use table in Software section 

message: I notice that there is currently no mention of a Riemann surface in the article.
Any multivalued analytic function, like the inverse function f-1 of a nonlinear analytic function f, has an interesting Riemann surface — call it M — which may be completed around finite branch points (those like the inverse function of z^n, n ≥ 2). On that Riemann surface (call it M') the inverse function is a bona fide, single-valued function, which (if there are any poles) can be taken further to a meromorphic completion, where the codomain ℂ ∪ {∞} now includes infinity, taking the topology of the sphere S2. Call the twice-completed surface M^.
The "completed" inverse function — call it by the same name f-1 — then has become a single-valued holomorphic function from the Riemann surface to the sphere: 
:::f-1 : M^  →  S2.
Of course the Riemann surface is the result of glueing all the branches together along their common edges. But it is also valuable to understand the Riemann surface explicitly. Especially because it has a topology and a conformal structure that are global properties, not merely local ones, that are essential invariants of the inverse function f-1 (now a bona fide single-valued meromorphic function).
If there is anyone reading this who is knowledgeable about the Riemann surface of the Lambert W function, I hope they will add it to the article. 2601:200:C000:1A0:C0A5:7B4A:921E:7CA (talk) 21:22, 4 July 2022 (UTC)
thread_title:  Riemann surface?