Talk:Lambert W function

Comment
For instance, to solve the equation	$2^{t}\,\!$	=	$5t\,\!$
we divide by $2^{t}\,\!$ to get	$1\,\!$	=	${\frac {5t}{2^{t}}}\,\!$
convert to exponential	$1\,\!$	=	$5t\,e^{-t\log 2}\,\!$
divide by 5	${\frac {1}{5}}\,\!$	=	$t\,e^{-t\log 2}\,\!$
multiply by -1 * log 2	${\frac {-\,log2}{5}}\,\!$	=	$-\,t\,log2\,e^{-t\log 2}\,\!$
replace $-t\log 2\,\!$ with $X\,\!$	${\frac {-\,log2}{5}}\,\!$	=	$X\,e^{X}\,\!$
Now application of the W function yields	$X\,\!$	=	$W\left({\frac {-\,log2}{5}}\right)\,\!$
replace $X\,\!$ with $-t\log 2\,\!$	$-t\log 2\,\!$	=	$W\left({\frac {-\,log2}{5}}\right)\,\!$
Isolate t	$t\,\!$	=	${\frac {-\,W\left({\frac {-\,log2}{5}}\right)}{log2}}\,\!$

Request

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)

Followup: Rob Corless has one (here: http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/expal2.1200.jpg), with accompanying explanation here: (http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/) maybe we should ask him? -- Tryptographer Thu Jan 8 04:39:47 EST 2009 —Preceding undated comment was added at 09:40, 8 January 2009 (UTC).

Python Algorithm

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)

Generalization of Lambert W function

The standard Lambert W function expresses exact solutions to what is called ``transcendental algebraic equations of the form:

exp(-c*x) = a_o*(x-r) where a_o, c, and r are real constants. (1)

The solution is: x = r + W(c*exp(-c*r)/a_o)

--> There has been a generalization of the Lambert W function[AAECC] within:

(i) a connection between gravity theory and quantum mechanics shown in the Journal of Classical

   and Quantum Gravity [gravity/QM]  where the RHS of (1) is now a quadratic polynomial:

   exp(-c*x) = b_o*(x-r_1)*(x-r_2)                               (2)

   where b_o, c, and r_1 and r_2 are real constants.

When r_1 = r_2, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function.

(ii) analytical solutions for the eigenenergies of a special case of the quantum 3-body problem,

    namely the hydrogen molecular ion.  Here the RHS of (2) is now an infinite order polynomial:

    exp(-c*x) = b_o* product_{i=1)^{infinity} (x-r_i)            (3)

    where b_o, c, and r_i for all 'i' are real constants.

Thus, it turns out that the Lambert W is of even greater fundamental importance than anybody realized.

--> I submit the current page on the Lambert function is INCOMPLETE and NEEDS TO BE UPDATED.

References (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)

Generalization

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)

Re: generalization.

Thanks. I hope the following clarifications answers your question(s):

Actually, it's one argument 'x'. The coefficients a_o, r_1, r_2 are parameters. This is similar to the hypergeometric and Meijer G functions which have only one argument but many parameters (as many as you like). This is implied by eq. (14) of the AAECC paper. I admit that the work of Scott et al. does not address everything. In particular the variation and dependencies on these parameters needs to be explored.

The paper by Scott et al. identifies the generalization and its governing equation(s) and provides exact solutions to some special cases but did not examine e.g. what happens when two distinct parameters are allowed to vary until they are no longer distinct. Nor did they examine this function in the complex plane. For eq. (2) both tasks were studied by Byers-Brown in the 1970s in a quantum chemistry journal for the the double well Dirac delta potential model. The resulting branch structure in the complex plane for eq. (2) is by no means trivial - harder IMHO than the average case for the hypergeometric functions. There were controversies about the convergences of the series not resolved until the 1990s by Scott, Dalgarno, Babb and Morgan. BTW, this body of work is cited in the references by Scott et al in the AAECC paper.

So a paper fleshing out e.g. series and asymptotic series expansions requires a whole body of work and paper in itself, even if some aspects have been worked out already...

But we're following the history of the standard W function. Firstly, one identifies the function or its structure (somehow). In this case, the governing equations are transcendental-algebraic equations which are now easy to solve numerically on most computers. What's missing from a practical point of view would be e.g. asymptotic series expansions for large argument 'x'. For the standard W function, that came relatively late in its history. Feedback is important. Wikipedia plays a role in this regard by helping this body of work connect to other people 'discovering' or re-discovering the Lambert W function in its generality.

PS: Since the generalization is there, shouldn't we edit out the part of the talk discussion which shows the generalization? —Preceding unsigned comment added by 171.71.55.135 (talk) 20:48, 11 December 2008 (UTC)

Need help simplifying

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)

See the equation $Y=Xe^{X}\;\Longleftrightarrow \;X=W(Y)$ in the article: $Y=-5e^{-5}$ means $W(Y)=-5$. Shreevatsa (talk) 22:04, 20 May 2009 (UTC)

Thanks but apparently there is a second solution. Try:

http://www58.wolframalpha.com and input ProductLog[-5*e^-5]

I am trying to understand where the second solution comes from.

Neil Parker (talk) 07:28, 21 May 2009 (UTC)

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

p^(ax+b)=cx+d

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)

Reference

From the article:

However the inverse of we^w 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)

DESY Lambert W

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)

DLMF notation

I suggest reverting [1], 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 authorative, 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 $\lambda =ae^{-\lambda }$, 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)