Talk:Polylogarithm

message: Can we follow the convention of force-rendering indented equations using \, not \!. People who don't want rendered equations will get their wish if their preference is set to "HTML if possible or else PNG" but with the stronger \! force rendering, they are stuck. \, will render the equation as PNG when the "HTML if very simple or else PNG" preference is set, for those of us who want PNG.
:Agreed.  I will do this from now on.  - Gauge 16:54, 20 May 2005 (UTC)
replies: text: Looks good to me. PAR 21:12, 19 Jun 2005 (UTC)
thread_title: Force rendering

message: This statement at the start of the third paragraph
"The special case s = 1 is the ordinary logarithm"
seems to say:
Li[1](z) = Ln(z)
In contrast to the first image under 
http://en.wikipedia.org/wiki/Polylogarithm#Particular_values
Which says:
Li[1](z) = -Ln(1-z)
Ac44ck 22:32, 28 August 2006 (UTC)
replies: text: Ok its fixed. PAR 23:49, 28 August 2006 (UTC)
thread_title:  Clarification? 

message: Arrghh. The formulae for Li_s(z) + (-)^s Li_s(1/z) are at variance with the definition of the principal 
branch of the complex logarithm given earlier on: -pi < Im(ln z) <= pi. The constraint accompanying the preceding 
formula for Li_s(e^(2 pi i x)) + (-)^s Li_s(e^(-2 pi i x)) clearly requires 0<x<1 for the second zeta argument. 
How about some experimentation for Im(z)<0 using a CAS? 145.254.103.78 (talk) 16:58, 17 April 2008 (UTC)
thread_title:  Inversion Formula Corrections 

message: Hi, 
I just posted a paper An efficient algorithm for computing the polylogarithm and the Hurwitz zeta functions (11 pages) with the following abstract:
: This paper develops an extension of the techniques given by Borwein's paper  "An efficient algorithm for computing the Riemann zeta function", to the polylogarithm and the Hurwitz zeta function. The algorithm provides a rapid means of evaluating         Lis(z) for general values of complex s and the region of complex z values given by  |z2/(z-1)|<3.3. This region includes the the Hurwitz zeta ζ(s,q) for general complex s and real 1/4≤ q ≤3/4. By using the duplication formula, the range of convergence for the Hurwitz zeta can be extended to the whole real interval 0<q<1, although the algorithm does run logarithmically slower as it approaches the endpoints. In particular, this algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable.
Comments/criticisms/corrections solicited on my talk page. linas 05:57, 28 December 2006 (UTC)
thread_title:  An efficient algorithm for computing the polylogarithm and the Hurwitz zeta functions 

message: While the Pentagon Identity given in terms of the dilogarithm Li_2 
holds for all arguments (x,y), the version
  L(x) + L(y) - L(xy) = L((x-xy)/(1-xy)) + L((y-xy)/(1-xy))
given in terms of the function
  L(x) := Li_2(x) + 1/2 ln(1-x) ln(x)
doesn't hold in general; it fails for instance with (x,y) = (2,-1), 
or (-i,-i), or (7/5 + i/3, 4/3 - i/2).
145.254.102.99 (talk) 14:23, 30 May 2008 (UTC)
thread_title:  Pentagon identity problems 

message: While reviewing this article, I note that the duplication formula
:\operatorname{Li}_s(z)+\operatorname{Li}_s(-z)= 2^{1-s}\operatorname{Li}_s(z^2)
is notable in its abscence. Any particuar reason?  Erm, never mind, why, there it is. I must be going blind.
FWIW the general multiplication formula is the Gauss sum, for integer p:
:\sum_{m=0}^{p-1}\operatorname{Li}_s\left(ze^{i2\pi m/p}\right)=p^{1-s} \operatorname{Li}_s(z^p)
Right?. linas 22:33, 29 December 2006 (UTC)
thread_title:  Duplication formula? 

message: This page launches straight in like a section in a maths textbook and assumes the reader is familiar with formal mathematical jargon and the general context of polylogarithms. This should not be assumed. I suggest adding a leading paragraph for nontechnical readers looking to understand the basics of what a polylogarithm is, and perhaps some of its more important applications in other disciplines, if any, per Wikipedia:Make_technical_articles_accessible#Articles_that_are_unavoidably_technical. It looks to me like it could be something like the analog of a logarithm in complex analysis, but I could be way off as the maths is a bit over my head. --Rogerb67 (talk) 11:13, 3 October 2008 (UTC)
replies: text: Changed  to , which seems more appropriate.--Rogerb67 (talk) 22:29, 3 October 2008 (UTC)
replies: text: I honestly don't think that a non-technical reader can hope to understand what a polylogarithm is. It might be possible, however, to communicate some basic idea of what a special function is (without simple redirection to that article) and the idea that this is one that comes up in problems with spin statistics (and another sentence on what those equally intractable things might be) (c.f. existing comments on B-E and F-D distributions). Comments on this opinion? One way or another, it's time to do something and get rid of that "fix me" box. As long as the introduction is under discussion, I also think that the pictures break up the text unnecessarily and detrimentally; shall we move these either below the introductory paragraphs or off to the side? Calavicci (talk) 20:13, 28 October 2009 (UTC)
replies: text:  I really don't know how to convey to a casual non-technical reader what a polylogarithm is. The simplest definition is the series expression at the top of the page. To paraphrase Forrest Gump, a polylogarithm is as a polylogarithm does. And what it does is to make those problems in BE and FD physics more "tractable". Even more, the BE distribution pops up in a non-physics way in graph theory, e.g. dynamic connectivity in the internet. And these are only the practical applications in my experience. I bet there are other practical applications, and there are loads of purely mathematical applications. Check the "what links here" link on the main page. There must be fifty links, and probably 80 percent are mathematical. How do you explain vector calculus to someone who has not yet learned algebra? You try to explain the problems it solves, but if the problems are beyond the experience of that person, you still have not shed any light on the subject. I think the best you can do is to say "study physics, study statistical physics, study quantum statistical physics, and then you will see the value of the polylogarithm and how it is used. I mean, it sounds like you have ideas on how the article should be but not the technical knowledge. I might have the technical knowledge, but no idea how to improve the article. If you want to go back and forth on this, maybe we can fix it. PAR (talk) 05:05, 29 October 2009 (UTC)
thread_title: Technical

message: I believe that the formula:
\lim_{\mathrm{Re}(\mu) \rightarrow \infty} \operatorname{Li}_s(e^\mu) = -{\mu^s \over \Gamma(s+1)}
~~~~~~(s\ne -1, -2,-3,\ldots)
 is incorrect, as written. Looking at the cited Wood reference, it seems like the given formula matches the expression given by Wood in Section 22...however, the associated text cites Equation 11.3, which has a negative sign (i.e. -e^mu). Also, Li(z)>0 for z>0, so a negative limiting behavior does not make sense. Finally, looking at a plot of x^(1/2)*Li_1/2(-e^(1/x)) using Mathematica suggests that the value of the function approaches -2/Sqrt[Pi] (or -1/Gamma[3/2]) as x->0+ (though there are admittedly significant numerical problems), which appears to lend credence to Wood's Equation 11.3. Thus, I suspect the correct formula is: 
\lim_{\mathrm{Re}(\mu) \rightarrow \infty} \operatorname{Li}_s(-e^\mu) = -{\mu^s \over \Gamma(s+1)}
~~~~~~(s\ne -1, -2,-3,\ldots)
. If noone objects/disagrees, I will fix this within the next couple of days. --GregRM (talk) 05:30, 2 December 2008 (UTC)
replies: text: I made the change, as no one seems to have objected or otherwise commented. For what it's worth, however, it looks like the original expression without the negative sign may be correct if we only consider the real part of the result.--GregRM (talk) 18:52, 7 December 2008 (UTC)
thread_title:  Suspected typo in limiting behavior 

message: It seems to me that the "Debeye integral" shown should have the following corrections in order for the subsequent series to work; Li_n, Z_n:
1>z>0 
/* Added 1>z on 9/10/2013 RR */
/*for z<0: then we need a term exp(z) in the numerator.
I retract this as the situation seems a lot more complicated RR
*/
I would make the corrections, but 217.184.57.149 seems to be putting in an effort to keep the page organized and correct.
Rrogers314 (talk) 16:18, 2 September 2013 (UTC)
replies: text: Hello. Could you be more explicit and also say where you differ from the referenced §16 of Wood? Also note the earlier discussion under "Errors in relations with Debye functions" above. A somewhat unusual definition of "Debye function" is used here, and I remember that I convinced myself that the expressions in the article are correct for this definition. 217.184.240.218 (talk) 21:09, 16 September 2013 (UTC)

text: You may be worried about the integral definition of Zn(z) in view of the integrand poles at t = 2k·π·i. Without explanation what path to follow from t=z to +infinity this definition cannot immediately be applied to the left half of the complex z plane. However, for n=1 Wood gives Z1(z) = −ln(1−exp(−z)) as the full definition, and more generally he proposes to calculate Zn(z) by means of the finite sum over polylogarithms of positive integer order given in the article. This introduces branch cuts from each of the points z = 2k·π·i to −infinity, and the integration path should be placed accordingly. 217.184.98.171 (talk) 19:36, 17 September 2013 (UTC)

text: Thanks for the response-- the requirement that z>0 for the 2nd Debye function is in A&S and evident from the series expansion there. I am rusty on the complex integration but you seem to imply there is a branch cut on  0<t<1 ? The z<1 was a mental flub; I meant that making z>0 would make the e^(-z) <1. BTW: Since the conversion coefficients are exp(u* ~H) (~H being the "shift matrix" from "The Matrices of Pascal and Other Greats") a great variety of other relationships are available. But I don't see much use in them. Drop me a talk line if your interested.Rrogers314 (talk) 18:33, 7 October 2013 (UTC)
replies: text: FYI, there are multiple branch cuts, making integration on the complex plan confusing and tricky. To add to the confusion, the second cut is not visible on the first sheet. 67.198.37.16 (talk) 17:09, 1 June 2021 (UTC)
thread_title:  In terms of the incomplete zeta functions or "Debye functions" : -Correction? 

message: wouldn't it be better to state this equation from the properties section:
:\textrm{Im}\left( \operatorname{Li}_s(z) \right) = -{{\pi \mu^{s-1}}\over{\Gamma(s)}} \,.
as
:\textrm{Im}\left( \operatorname{Li}_s(e^{\mu}) \right) = -{{\pi \mu^{s-1}}\over{\Gamma(s)}} \,.
It wood match Wood's notation and the hint to the logarithm wouldn't have to be searched in the text....
88.217.62.143 (talk) 20:38, 3 August 2014 (UTC)
replies: text: No, because that would be incorrect -- there would be an ambiguity of 2\pi n. The text makes clear that \mu is for the principle branch. The subtlety is that the polylogarithm contains two branch cuts, not one, and thus has a non-trivial monodromy. 67.198.37.16 (talk) 17:05, 1 June 2021 (UTC)
thread_title: Clarification

message: In the limiting behavior section, I believe the equation for |\mu| \to 0 gives the correct answer. However the reference that is cited (Woods), gives \lim_{|\mu| \to 0} \mathrm{Li}_s \! \left( e^{\mu} \right) = \Gamma \left( 1 - s \right) \left( -\mu \right)^{s}. Is there a way to verify this? Another reference would probably be best, but the discrepancy should at least be noted.  — Preceding unsigned comment added by 2607:EA00:107:3C01:B445:1766:B090:FB9C (talk) 21:48, 2 November 2016 (UTC)
replies: text: Huh? The article currently gives the formula you give above. The article as it stood on 2 November 2016 also gave this same formula. So I can't tell what you are complaining about. 67.198.37.16 (talk) 17:00, 1 June 2021 (UTC)

text: The discrepancy is only in the section near the end labeled Limits. The equation Wood cites is (9.3) which has  the exponent s-1. See https://www.cs.kent.ac.uk/pubs/1992/110/content.pdf Eq. 9.3 and 9.6. 128.138.141.236 (talk) 23:53, 4 May 2022 (UTC)
thread_title:  An error in a reference? 

message: The formula right below "Introducing an explicit expression for the Stirling numbers of the second kind into the finite sum for the polylogarithm of nonpositive integer order (see above) one may write" doesn't work for me. I suspect it is wrong because it is very close to the explicit formula for an Eulerian number (see https://en.wikipedia.org/wiki/Eulerian_number), but subtly different. In particular, the term of the Combinations term is different.  — Preceding unsigned comment added by Blaisegassend (talk • contribs) 00:16, 7 May 2017 (UTC)
thread_title:  An error in a formula?