Talk:Polylogarithm

Force rendering

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)

Is there a consensus on how to handle bulleted lists like the ones found on this page? It seems the wiki style and html style have different indentation properties, and I was wondering if one was preferred over the other here. I only noticed this after today's edit, so feel free to change them back if you liked the previous style better. - Gauge 18:59, 19 Jun 2005 (UTC)

Looks good to me. PAR 21:12, 19 Jun 2005 (UTC)

Borwein reference

Paul asks Linas:

With respect to your recent addition to the Polylogarithm article, I was wondering which Borwein reference you were using (there are two listed). Also is there an easy way to see how the two sums are equal? Thanks - PAR 8 July 2005 18:47 (UTC)

The ref I pulled this out of is actually Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". J. Comp. App. Math. 121: p.11. {{cite journal}}: |pages= has extra text (help); External link in |title= (help)CS1 maint: multiple names: authors list (link) The above ref barely mentions polylogs, so I'm not sure its a worthwhile ref for this article. I did not verify this particular formula directly, just now, although I have derived a whole zoo that are very similar in nature in the past (this one looked correct and seemed worth jotting down). The trick for deriving these that I like to use is exploring the self-similarities of fractals; one gets entire rafts of these, for general rational numbers. The crazy rational-number relations on the Hurwitz zeta function is the source of these. The poles correspond to eigenvalues of transfer operators.

Off-topic: have you seen any good compendiums on the Lerch zeta function? I need some identities, and it seems that whenever I derive them manually, with great pain, I invariably find them in a book, a few weeks later. :( I need sums of the form

$\sum _{n=-\infty }^{\infty }\exp(2\pi inx)/(n+a)^{s}$

for s=1. These show up as related to the eigenfunctions of the ideal that generates the Weyl algebra. -- linas 8 July 2005 19:19 (UTC)

A very good reference is The Lerch Zeta-function by Laurincikas and Garunkstis (spelled without the diacritical marks.) This is not to say I understand its contents completely, but I might be able to find a formula for you. PAR 9 July 2005 04:01 (UTC)

Also - I saw you changed back to $e^{\mu }=1$ from $\mu =0$. I saw your original change to $\mu =0$ and thought it was an error too, until I looked at the beginning of the article to see that $\mu$ is defined to be the principle value of the logarithm, so $\mu =0$ is correct. By the way, I think Ln(x) is the general multi-valued logarithm and ln(x) is the principal value, which is the reverse of what is in the article. It that correct? If so I will change it. PAR 9 July 2005 04:37 (UTC)

Yes. According to Abramowitz and Stegun, Ln(s) is the multi-valued function and ln(s) is the principle sheet.

Also ... maybe its worth splitting this article into two? It takes a while to load in the browser.

As to the fixes: I was not looking at the top of the article, but only locally: there was no z in the nearby formulas, so I assumed z was a typo. Next, the sum over poles clearly fails for μ=0, so I made a change to say that. Then silly me, I note that the sum fails for all μ=2πin for any integer n. Well, I could have just written that, but then I thought, what is the easiest way to say that this fails? Answer: when $e^{\mu }=1$, and so that was the change I made. I was less concerned about global consistency over the entire article (which is important), but about local consistency 9which is even more important). -- linas 9 July 2005 17:08 (UTC)

Hi Linas - I agree, I think the change you made is best. I exchanged ln and Ln in the article, and now I'm suspicious of the equations in the Polylogarithm#Particular values section. This is the only place that ln occurred before. I will check them out, but if you get a chance, could you do the same? Thanks PAR 19:40, 9 July 2005 (UTC)

PS - I'm against splitting the article. I assume its slow because you are on dialup? People with broadband will be disadvantaged by splitting it. Since the direction is towards more broadband, I say leave it.

Error??

I think I see an error in one section, which is perpetuated. Please review. In the section "Series representations", there is a nice derivation, which starts as follows:

We may represent the polylogarithm as a power series about μ = 0 as follows: Consider the Mellin transform:

$(1)\,\,\,M_{s}(r)=\int _{0}^{\infty }{\textrm {Li}}_{s}(fe^{-u})u^{r-1}\,du={1 \over \Gamma (s)}\int _{0}^{\infty }\int _{0}^{\infty }{t^{s-1}u^{r-1} \over e^{t+u}/f-1}~dt~du.$

Above looks good to me.

The change of variables t = ab, u = a(1 - b) allows the integrals to be separated:

$(2)\,\,\,M_{s}(r)={1 \over \Gamma (s)}\int _{0}^{1}b^{r-1}(1-b)^{s-1}db\int _{0}^{\infty }{a^{s+r-1} \over e^{a}/f-1}da=\Gamma (r){\textrm {Li}}_{s+r}(f).$

The first integral is the Beta function, and this is where the mistake is made. I get the following:

$(3)\,\,\,B(s,r)=\int _{0}^{1}b^{r-1}(1-b)^{s-1}db={\frac {\Gamma (r)\Gamma (s)}{\Gamma (r+s)}}$

and so then I get

$(4)\,\,\,M_{s}(r)={\frac {\Gamma (r){\textrm {Li}}_{s+r}(f)}{\Gamma (r+s)}}$

which is not what's in the article. Or am I hallucinating? The article continues:

For f = 1 we have, through the inverse Mellin transform:

$(5)\,\,\,\operatorname {Li} _{s}(e^{-u})={1 \over 2\pi i}\int _{c-i\infty }^{c+i\infty }\Gamma (r)\zeta (s+r)u^{-r}dr$

But this should now read:

$(6)\,\,\,\operatorname {Li} _{s}(e^{-u})={1 \over 2\pi i}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (r)}{\Gamma (r+s)}}\zeta (s+r)u^{-r}dr$

Right? This missing factor seems to be perpetuated down the line. Please review, let me know. linas 14:07, 2 May 2006 (UTC)

---

Equation 4 should read:

$M_{s}(r)=\left({\frac {1}{\Gamma (s)}}\right)B(s,r)\left(\Gamma (s+r)\operatorname {Li} _{s+r}(f)\right)=\Gamma (r)\operatorname {Li} _{s+r}(f)$

I think you forgot that the second integral in the middle part of equation 2 is not the polylogarithm, but $\Gamma (s+r)$ times the polylogarithm. I have not checked past that point. I am not an error-free person, so let me know if this looks right. PAR 17:04, 2 May 2006 (UTC)

Dohhh. Thank you. I'm embarrassed. linas 00:19, 3 May 2006 (UTC)

Clarification?

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://wiki.riteme.site/wiki/Polylogarithm#Particular_values

Which says: Li[1](z) = -Ln(1-z)

Ac44ck 22:32, 28 August 2006 (UTC)

Ok its fixed. PAR 23:49, 28 August 2006 (UTC)

Confusion

The article currently states:

The polylogarithm is related to the Hurwitz zeta function by:

$\operatorname {Li} _{s}(e^{2\pi ix})+(-1)^{s}\operatorname {Li} _{s}(e^{-2\pi ix})={(2\pi i)^{s} \over \Gamma (s)}~\zeta \left(1-s,x\right)$

where Γ(s) is the gamma function. This holds for

${\textrm {Re}}(s)>1,{\textrm {Im}}(x)\geq 0,0\leq {\textrm {Re}}(x)<1$

and also for

${\textrm {Re}}(s)>1,{\textrm {Im}}(x)\leq 0,0<{\textrm {Re}}(x)\leq 1.$

(Note that Erdélyi's equivalent Equation [Erdélyi 1981 § 1.11-16] is not correct if we assume that the principal branches of the polylogarithm and the logarithm are used simultaneously.) This equation furnishes the analytical continuation of the series representation of the polylogarithm beyond its circle of convergence|z|= 1.

Alternatively, for all $s\in \mathbb {C}$ and for all $z~\not \in ~]1;+\infty [$, the inversion formula is

$\operatorname {Li} _{s}(z)+(-1)^{s}\operatorname {Li} _{s}(1/z)={(2\pi i)^{s} \over \Gamma (s)}~\zeta \left(1\!-\!s,{\log z \over 2i\pi }\right),$

I've several comments/questions:

• The restriction to ${\textrm {Re}}(s)>1$ for the first part seems directly contradicted by for all $s\in \mathbb {C}$ in the second part. There's not anything subtle I'm missing here, is there?
• What exactly is the notation $z~\not \in ~]1;+\infty [$ supposed to mean? The reversed brackets confuse me; should I read this as $z~\not \in ~[1;+\infty ]$? Are the reversed brackets supposed to denote "principle branch", perhaps?
• The logarithm has a branch cut that extends from 0 to the left, so we should also say $z~\not \in ~[-\infty ;0]$, to be pedantic, right?

Thanks linas 03:15, 14 October 2006 (UTC)

The reversed brackets are commonly used to denote open intervals in European literature (and standardized in ISO 31-11). And no, I don't like them. Fredrik Johansson 07:09, 14 October 2006 (UTC)

This entry should be rewritten and checked for accuracy. For one, it does not follow the convention of using ln for the principle branch of the natural logarithm. Also, I am used to () to express open intervals, and I don't know what is standard on Wikipedia, but whatever it is, it would be preferred. These additions were made by anonymous 80.203.48.14 and 128.39.229.124 in November of 2005 and are probably the same person - both users contributions consist entirely of their polylogarithm edits. PAR 13:41, 14 October 2006 (UTC)

regarding the first objection: The restriction to Re(s) > 1 is unnecessary and has been removed. Additionally, Im(x)≥ 0 has been corrected to Im(x) > 0.

regarding the third objection: The logarithm on the negative real axis is assumed to belong to the upper half plane, -π < Im(ln z) ≤ π, and z in [-∞,0] is allowed. 62.180.184.50 (talk) 13:39, 14 April 2009 (UTC)

I am changing the constraint from the present "0 < Re(x) <= 1 if Im(x) <= 0, and 0 <= Re(x) < 1 if Im(x) > 0" to "0 <= Re(x) < 1 if Im(x) >= 0 , and 0 < Re(x) <= 1 if Im(x) < 0". This excludes x=1 and replaces it by x=0. Since zeta(1-s,0) = zeta(1-s,1) for Re(s)>1, this is of relevance only for Re(s)<1, where now the x=0 pole of zeta(1-s,x) rather than the regular value at x=1 is related to the z=1 poles of Li_s(z) and Li_s(1/z). Correspondingly, I am replacing the excluded interval "]0;1[" by "]0;1]". 62.180.184.13 (talk) 20:45, 5 September 2009 (UTC)

Inversion Formula Corrections

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)

Made the necessary corrections. 145.254.104.227 (talk) 19:15, 29 April 2008 (UTC)

More on the inversion formulae

To editor 85.164.137.141 - I reverted your edits not because they were wrong, but because I am not sure how to get your attention. I notice that these three edits are the only edits you have made on wikipedia, so I am not sure if you understand User talk pages or even article talk pages.

If you are reading this, could you please supply a reference or an explanation for your edits to the inversion formula? Thanks. PAR 01:19, 29 October 2006 (UTC)

I am the author of the general inversion formula and of their corrections. Indeed, I'm not familiar with the use of Wikipedia. I apologize if I created troubles; that was not my intention.

Concerning the general inversion formula:

• I do not know any reference where one can find them (I don't have access to all the literature on the subject); such a reference probably exits, though. I derived them myself (not rigorously, I must admit) and checked numerically against Maple 10.
• The apparent contradiction with the other inversion formula restricted to ${\textrm {Re}}(s)>1$ is probably due to the fact that they were (I guess) derived from the integral definition:

$\operatorname {Li} _{s}z\ =\ {1 \over \Gamma (s)}\int _{0}^{\infty }{t^{s-1} \over \mathrm {e} ^{t}/z-1}\,\mathrm {d} t$

which is valid for ${\textrm {Re}}(s)>1$.

• A general definition valid $\forall z$ and $\forall s$ derives directly from the definition of the Lerch transcendent:

$\operatorname {Li} _{s}z\ =\ {z \over 2}\ +\ (-\log z)^{s-1}\,\Gamma (1-s,-\log z)\ +\ 2z\int _{0}^{\infty }{\frac {\sin(s\arctan t\,-\,t\log z)}{(1+t^{2})^{s/2}\,(\mathrm {e} ^{2\pi t}-1)}}\,\mathrm {d} t$

where $\log$ is the principal branch of the logarithm and $\Gamma$ is the incomplete Gamma-function. This definition should allow rigorous derivations of various relations with a more precise validity range in the variables $z$ and $s$. Note that, in this expression, all (but not part) of the $\log(z)$ can be replaced by $-\log(1/z)$. Note also that, although this definition seems compatible with Maple's one (checked numerically), it is quite possible that other programs/authors use different definitions.

Concerning the notations:

• Abramowitz & Stegun ones are not universal. I am more used to denote the natural (Naperian) logarithm "ln" and the principal branch of the complex logarithm "log". Some Wikipedia pages use this convention.
• As a matter of personal taste, I prefer to denote open intervals with reversed squared bracket because, e.g., $(0,1)$ can be interpreted as the coordinates of one point in the plane and not as the open interval $]0,1[$.
• These considerations are very secondary as long as the notations are clearly explained.

Finally, I think that this Wiki page is getting a little messy, and one could clean it up. I am not the most qualified person to do it, and I do not want to mess with the inputs from more competent contributors. Also, for the same reasons, I didn't re-revert the corrected inversion formula. —The preceding unsigned comment was added by 84.48.121.237 (talk • contribs) 12:46, 3 December 2006 (UTC).

An efficient algorithm for computing the polylogarithm and the Hurwitz zeta functions

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 Li_s(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)

Pentagon identity problems

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).

The offending material has been removed.145.254.104.117 (talk) —Preceding comment was added at 19:31, 8 June 2008 (UTC)

Duplication formula?

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)

Technical

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)

Changed {{technical}} to {{context}}, which seems more appropriate.--Rogerb67 (talk) 22:29, 3 October 2008 (UTC)

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)

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)

I am removing the fairly recent "Too Technical" sticker to get the attention of the person responsible.

As the preceding comments show, this issue has been raised before. In fact the level of this article is not untypical of the treatment of a special function of higher mathematics in Wikipedia: it simply cannot be understood without a good knowledge of mathematical functions and complex analysis. However, everybody else (surely the vast majority of Wikipedia users) can and should try to follow – if necessary, recursively – the numerous links to articles on the more basic subjects, in order to get a detailed idea of the background.

If you have specific criticisms of the technical level of a particular sections of this Wikipedia article, it would be very helpful if you could express them here, so that the specialists may consider what should be done about them. 217.184.241.158 (talk) 08:31, 17 February 2013 (UTC).

Suspected typo in limiting behavior

I believe that the formula:$\lim _{\mathrm {Re} (\mu )\rightarrow \infty }\operatorname {Li} _{s}(e^{\mu })=-{\mu ^{s} \over \Gamma (s+1)}~~~~~~(s\neq -1,-2,-3,\ldots )$ 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: $\lim _{\mathrm {Re} (\mu )\rightarrow \infty }\operatorname {Li} _{s}(-e^{\mu })=-{\mu ^{s} \over \Gamma (s+1)}~~~~~~(s\neq -1,-2,-3,\ldots )$. If noone objects/disagrees, I will fix this within the next couple of days. --GregRM (talk) 05:30, 2 December 2008 (UTC)

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)

Errors in relations with Debye functions

Both formulas expressing Debye functions expressed in terms of polylogarithms and vice versa are incorrect. These are the last two formulas in the section "Relationship to other functions". My suggestion concerning the last formula is the following:

$Z_{n}(\mu )\,=\,{\frac {n\,n!}{\mu ^{n}}}\zeta (n+1)\,-\,n\,n!\sum _{k=0}^{n}\operatorname {Li} _{k+1}(e^{-\mu }){\frac {\mu ^{-k}}{(n-k)!}}\,,\quad n=1,2,3,\ldots$

This formula has been derived by successive partial integrations of the Debye integral and been tested numerically using MATHEMATICA. The correction of the first formula has still to be done. [This is my first Wikipedia talk, forgive me if something is wrong..] Heinzjuergen (talk) 18:26, 7 May 2009 (UTC)

So the cited reference Abramowitz & Stegun § 27.1, 27.7.7 is also wrong? Or does it actually say something different from Wikipedia at present? 62.180.184.6 (talk) 22:25, 24 August 2009 (UTC)

I have looked into the matter. Provided that the "Debye functions" are defined by

Z_n(z) = 1/(n-1)! int_z^infty t^(n-1)/(e^t-1) dt, (n = 1,2,3,...)

(which doesn't agree with the function D_n defined at the link provided in the article), the relations in the article are correct. I am therefore supplying this definition of Z_n in the article along with the two relations.

By the way, there is a similar situation with the "Inverse tangent integral" Ti_s(z) earlier on, where the link also gives no clue what is meant: one is redirected back to the polylogarithm page! 62.180.184.57 (talk) 08:19, 11 September 2009 (UTC)

Edits of 22 August 2009 by Sajoka

It is a good idea to read an article before making changes to it. Sajoka's additions to the "Particular values" Section paraphrase what is already found in the "Relationship to other functions" Section, where the matter indeed belongs. Thus Sajoka's introduction of

  b(n,x) = -n! (Li_n(-e^(i x)) + (-1)^n Li_n(-e^(-i x)))

coincides with

  Li_n(z) + (-1)^n Li_n(1/z) = -(2 pi i)^n/n! B_n(1/2 - ln(-1/z)/(2 pi i))

from that Section if z is set to -e^(i x). And that Dirichlet's eta(2n) relates to Li_(2n)(-1), that Dirichlet's beta(2n+1) relates to Li_(2n+1)(+-i), and that Riemann's zeta(2n) relates to Li_(2n)(1), are also found in that Section in a much more general (hence more useful) form.

This is why I am removing Sajoka's additions. 62.180.184.6 (talk) 22:25, 24 August 2009 (UTC)

To Sajoka: your (modified) addition to the "Particular values" Section doesn't meet the objections expressed above:

• the addition doesn't concern particular values, but the relationship of the polylogarithm Li_s (for positive integer s) with the Hurwitz zeta function (i.e. the Bernoulli polynomials). It therefore belongs into the "Relationship to other functions" Section.

• as stated above, more general forms of the relationship with the Hurwitz zeta function are already given in the "Relationship to other functions" Section. The only difference is that you write the Bernoulli polynomial as a sum of powers. This particular topic belongs to the article on Bernoulli polynomials, however.

• at the start of the "Particular values" Section, the reader is already referred to the "Relationship to other functions" Section.

Both your modified addition and its placement are therefore still unsuitable; I will remove the material again in the near future. Please address my objections on this page if you like. 62.180.184.13 (talk) 18:47, 29 August 2009 (UTC)

Ok you are right, you may remove it. —Preceding unsigned comment added by Sajoka (talk • contribs) 14:56, 30 August 2009 (UTC)

Polylogarithm, 62.180.184.13 has smiled at you! Smiles promote WikiLove and hopefully this one has made your day better. Spread the WikiLove by smiling at someone else, whether it be someone you have had disagreements with in the past or a good friend. Go on, smile! Cheers, and happy editing!
Smile at others by adding {{subst:Smile}} to their talk page with a friendly message.

Thanks for your agreement! 62.180.184.13 (talk) 20:45, 5 September 2009 (UTC)

Footnotes

{{needs footnotes}} is the best notice here, even though its wording isn't quite adequate. Although you're using {{harv}}, there's a fair amount of text in the article where the cross-link from text to citation of supporting source is not supplied to the reader. Uncle G (talk) 10:49, 23 September 2010 (UTC)

I am the primary contributor to the polylogarithm article for the last two years or so (2009-2010) and I am familiar with the literature – printed (overview and historical papers from the reference list, a number of which were added by me) as well as on-line (Wolfram MathWorld and NIST dlmf) material. What caused me to work on this Wikipedia article was that it had a lot of substance already; by now it could be the most accurate and most complete resource available for the basic properties of the polylogarithm of complex order and complex argument – it was better than dlmf when dlmf appeared this year. I will perhaps work a bit more on the dilogarithm chapter and mention some of the history; otherwise I am convinced that more detailed citations are not practicable here.

One reason is that not many works cover the case of non-integer (and possibly complex) order and of complex argument; another is that some old mistakes are regularly perpetuated in the literature (for a recent example see the inversion relation (9.5) in the Maximon reference) – note that the present article contains a handful of warnings about this. Moreover, some basic formulae probably cannot be found anywhere in a form close to what appears here. Briefly put, in mathematics – where we are dealing with an unambiguous notion of truth – internal consistency and a self-explanatory exposition should take precedence over the assembly of an article from literature quotations (which may be wrong, or not apply in all circumstances).

Presumably in consequence of the "fix me" sticker pasted on top of the reference list, the Abramowitz & Stegun reference (a main mathematical reference work of the 20th century) has recently been deleted by a reader (indeed it contains no polylogarithm chapter); unfortunately, however, this reference is cited three times in the text (as A&S §27.8, §27.1 & 27.7.7, §27.7). Since no replacement citations were provided I am going to restore this reference. (A replacement for the second citation will be very hard to find, as its purpose is to justify the use of the name Debye function for a certain entity.) At the same time, I am going to remove the "fix me" sticker, as it is proving to be counterproductive – as far as I remember, no other reference was removed during the last two years, especially none that is actually cited in the text. 217.184.57.140 (talk) 11:07, 9 October 2010 (UTC)

The second part of the second citation mentioned above (i.e. A&S §27.7.7) has been replaced by a more satisfactory Wood §16. 217.184.57.149 (talk) 12:33, 9 October 2010 (UTC)

Polylogarithm inversion formula

I am interested in the "inversion formula" for this function, namely, with n in Z,

Li_n(z) + (-1)^n Li_n(1/z) = -(2 i pi)^n /n! B_n(1/2 + log(-z/(2 i pi))

in case z (in C) does not belong to [0, 1], etc., as given in the excellent polylogarithm article. There it is stated that the formula given in Erdélyi et al. (1981) is again not correct (also my experience in some formulae).

My question: I am interested in a reference to the proof of this corrected formula. The classical reference to the matter is the well-known, outdated Jonquière paper of 1889, with the wrong formula. — Preceding unsigned comment added by 79.235.232.78 (talk) 09:47, 25 April 2012 (UTC)

[I have edited your question a bit for readability, even though one should not do this!] The inversion formula for polylogarithms of integer order is just a special case of the inversion formula for arbitrary complex order s. As stated in the article, the latter is a consequence of the relation expressing the polylogarithm in terms of the Hurwitz zeta function: "The polylogarithm is consequently also related ...". In particular, by adding two instances of this relation, you get an inversion formula that involves four Hurwitz zeta functions. In reducing these to a single one, you may find it advantageous to omit the positive real axis at first, since you then have ln(−z) = −ln(−1/z). Arguments on the real axis can finally be handled by means of a well-known functional equation between ζ(s, a) and ζ(s, a+1).

You may also want to consult this paper by B.C. Berndt, which has the only proof freely available on the web that I am aware of at the moment (apart from the Jonquière and Wood references given in the article), but deals with the more general case of the Lerch transcendent. 217.184.105.53 (talk) 08:01, 17 May 2012 (UTC)

I have unscrambled some sentences in my own comment immediately above. 217.184.226.233 (talk) 19:11, 10 June 2012 (UTC)

I have added the explanatory phrase "With a little help from a functional equation for the Hurwitz zeta function ..." in the article. 217.184.240.120 (talk) 20:14, 17 May 2012 (UTC)

And I have included the Jonquière references where readers are warned that the formulae in Erdélyi at al. (1981) are not entirely correct. 217.184.224.2 (talk) 11:45, 28 May 2012 (UTC)

Erroneous special dilogarithm values

The last two special values of Li2 for arguments (√5+1)/2 and (√5+3)/2 are wrong since the Li2(z) is not real for z>1. I have no idea what the Li2 is for those arguments. I have numerically confirmed the other table entries. NormHardy (talk) 03:55, 28 August 2012 (UTC)

Sorry for not noticing your message earlier. I believe I assembled this table a few years back. Actually,

11/15 pi^2 + 1/2 ln(−(√5−1)/2)^2

and

−11/15 pi^2 − ln(−(√5−1)/2)^2

are complex rather than real because the above logarithm arguments are negative. And the numerical value of the first expression, 2.418690103 − 1.511771534 i, does indeed agree with that of Li2((√5+1)/2). But the value of the second expression, 2.400329686 + 3.023543068 i, only agrees with the complex conjugate of Li2((√5+3)/2). So the correct expression for Li2((√5+3)/2) will be

−11/15 pi^2 − ln(−(√5+1)/2)^2,

right? I will make the correction. Thanks for unearthing this one! 217.184.226.63 (talk) 02:59, 22 September 2012 (UTC)

In terms of the incomplete zeta functions or "Debye functions" : -Correction?

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)

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)

You may be worried about the integral definition of Z_n(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 Z_n(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)

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)

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)

Clarification

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)

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)

An error in a reference?

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)

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)

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)

An error in a formula?

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://wiki.riteme.site/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)

Limiting behavior

The way that these formulas are presented is incorrect, it makes no sense to write lim_x f(x) = g(x). I assume that these are asymptotic equivalences, so I checked Wood (1992) to verify this -- and also see whether some of these convergences might be uniform in the second variable. However, the results are also incorrectly stated in that reference. In addition to that, there seemingly are some problems in a few of Wood's formula (as reflected on this talk page). So, overall, I think the whole section is dubious and should be thoroughly checked. It would also be nice to have another, more reliable source. Malparti (talk) 23:26, 23 January 2021 (UTC)

You have mis-read the formulas. The are stating that lim_h(x) f(x) = g(x) where h(x) is the absolute value or the real part, etc. For example, the first formula

$\lim _{|z|\to 0}\operatorname {Li} _{s}(z)=z$

could be written with big-O notation as

$\operatorname {Li} _{s}(z)=z+{\mathcal {O}}(|z|^{2})$

(Effectively. I'm not quite sure, it might actually be something like this

$\operatorname {Li} _{s}(z)=z+{\mathcal {O}}(|z|^{2}\log |z|)$

or something else similar; it would require some work to check. The limit form conveys less information than the big-O notation.)

I'll therefor remove the nag note. 67.198.37.16 (talk) 16:48, 1 June 2021 (UTC)

Thanks for the reply. However I have not misread the formulas. As I wrote in my comment, the problem comes from the fact that these are asymptotic equivalents, not limits. For instance, the specific example you chose:

$\lim _{|z|\to 0}\operatorname {Li} _{s}(z)=z$

is not a correct mathematical expression and is ambiguous. Indeed, there are two ways to interpret it: as a limit

$\lim _{|z|\to 0}\operatorname {Li} _{s}(z)=0$,

or as an asymptotic equivalent

$\operatorname {Li} _{s}(z)\sim z$ as $|z|\to 0$,

which of course is a stronger statement. If one can quantify the speed of convergence with a O term, then that is even better.

I do not do a ton of complex analysis so that abuse of notation might be frequent in that field (I doubt it). It certainly isn't in mine, and it is the first time I see it in a published math article. At any rate, for a project such as Wikipedia I think it is better to use standard notation and to avoid notation that we explicitly tell uni students not to use, such as lim_h(x) f(x) = g(x).

As a result, I have put the "nag note" back -- not sure why you refer to it that way.

In case anyone else reads this and is interested in the second order term: it pretty clearly is $z^{2}/2^{s}$. ;)

Best, Malparti (talk) 08:54, 21 June 2021 (UTC)

I dunno. I've seen that kind of notation in books, maybe Hardy & Littlewood, for example, maybe Boas & Buck. Don't recall. Seems commonplace enough. Doesn't particularly bother me, the meaning is utterly unambiguous. It's just a short-hand for saying that you're keeping the radius constant in the limit, while the angle is completely unconstrained. But sure, someone should jog down to the library and actually look at the book. 67.198.37.16 (talk) 05:55, 22 June 2021 (UTC)

Sorry to insist, as this is only tangentially relevant to the article and I am mostly going to repeat myself. But this is relevant to the present discussion.

I do not think this notation is "commonplace enough". I read math for a living, and I have never seen this notation except in Wood's technical report (and of course in the writings of confused uni students). With all due respect, I think you have not understood what the problem is and where the ambiguity comes from, since this has nothing to do with the angle of approach. Writing $\lim _{z\to a}$ when the limit exists and is the same irrespective of the direction is fine and is, indeed, standard. By contrast, the notation $\lim _{z\to a}f(z)=g(z)$ is ambiguous: should it be interpreted as $\lim _{z\to a}f(z)-g(z)=0$, or as $\lim _{z\to a}f(z)/g(z)=1$? (here it is the second meaning that is assumed; confused students usually assume the first). Malparti (talk) 15:46, 23 June 2021 (UTC)

Please observe that the limits are written as $\lim _{|z|\to 0}$, and not $\lim _{z\to 0}$ or $\lim _{z\to a}$. This offers a big clue as to the intended interpretation, and yet this seems to be the source of confusion. I believe this is called a "directional limit" and is commonly used when something particular happens in one direction that does not happen in another. It's analogous to a partial derivative, or to the gradient in multi-variate calculus: the gradient depends on the direction you are moving in.

I don't see any ambiguity: both $\lim _{|z|\to 0}\operatorname {Li} _{s}(z)-z=0$ and $\lim _{|z|\to 0}\operatorname {Li} _{s}(z)/z=1$ are true statements, and both capture the intended meaning. This works because if $f(z)=z+{\mathcal {O}}(|z|^{n})$ then clearly $f(z)-z={\mathcal {O}}(|z|^{n})$ and the limit makes the ${\mathcal {O}}(|z|^{n})$ vanish when $n\geq 2$. Similarly, $f(z)/z=1+{\mathcal {O}}(|z|^{n-1})$ and the higher order term again vanishes (when $n\geq 2$). You are not going to get different results this way, which is why there is no ambiguity.

As to where this notation is used: perhaps it is more common in engineering and physics, than it is in pure mathematics. Even with modern search engines, there is no way to perform a search to see where else such notation is used. I can attest that I've seen it more than a few times, and its never been problematic or confusing. 67.198.37.16 (talk) 17:42, 13 July 2021 (UTC)

As to where this notation is used: I busted open my copy of Abramowitz and Stegun and see a very similar notation rampantly used, particularly for asymptotic limits. For example, eqn 6.3.18 is written as $\psi (z)\sim \ln z-1/2z+\cdots (z\to \infty {\mbox{ in }}|\arg z|<\pi )$ The tilde appears, instead of an equals sign, because it is an asymptotic limit. Rather than specifying the limit parenthetically, as A&S do, other authors just stick a "lim" in there, rather than using parenthesis. If it makes you happier, you can rewrite the existing

$\lim _{|z|\to 0}\operatorname {Li} _{s}(z)=z$

either as

$\operatorname {Li} _{s}(z)\sim z+\cdots \quad (|z|\to 0)$

or as

$\operatorname {Li} _{s}(z)=z+{\mathcal {O}}(|z|^{2})\quad (|z|\to 0)$

would that work? 67.198.37.16 (talk) 18:19, 13 July 2021 (UTC)

Yes either of those would be better (though I would write the first as $\operatorname {Li} _{s}(z)\sim z\quad (|z|\to 0)$ since the "+ \cdots" adds no information at all). (Of course the second one is only better if it's the correct error bound.) --JBL (talk) 19:32, 13 July 2021 (UTC)

Well, it feels like I'm being railroaded into making those changes, even though I never wrote that part of the article, and I'm not bothered by how it currently stands. If this really is the popular demand, I could hack it, but it's not my first choice of action. 67.198.37.16 (talk) 23:54, 15 July 2021 (UTC)

Please make those changes. I also find the current notation confusing.105.225.121.92 (talk) 06:15, 15 September 2021 (UTC)

This is just to say that just now, yet another user was — legitimately — confused by the horrendous and mathematically incorrect way things are written.

So I will try to fix things by systematically replacing all limits by asymptotic equivalents. However, I won't take the time to actually check that the formulas are accurate, and I still consider Wood (1992) to be an unreliable source. So someone with more interest than me in the polylogarithm should still take the time to do this. Malparti (talk) 17:01, 27 May 2024 (UTC)

More about asymptotic equivalents and limits

Casual reader here. I'm on board with recent changes in the "Limits" section, to write each item as an asymptotic equivalence. However, the result is a little confusing to me. (1) Can we make the title match the content of the list? Either the title being "Limits" and each item is stated as a limit of some kind, or the title being "Asymptotic equivalences" and each item is stated as an asymptotic equivalence. (2) If written as asymptotic equivalences, can we separate the "z -> something" from the big tilde? I find putting them together is cluttered. How about if we just write "Li[s](z) ~ f(z) as z -> something". Thanks for considering these thoughts. Wile E. Heresiarch (talk) 13:38, 12 September 2024 (UTC)

@Wile E. Heresiarch Sure! Feel free to implement the changes. Best, Malparti (talk) 07:45, 23 September 2024 (UTC)