Talk:Polylogarithm

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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)

Contents

[edit] 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.  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 i nx) / (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 eμ = 1 from μ = 0. I saw your original change to μ = 0 and thought it was an error too, until I looked at the beginning of the article to see that μ is defined to be the principle value of the logarithm, so μ = 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 eμ = 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.

[edit] 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 Γ(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)

[edit] 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://en.wikipedia.org/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)

[edit] Confusion

The article currently states:

The polylogarithm is related to the Hurwitz zeta function by:
\operatorname{Li}_s(e^{2\pi i x})+(-1)^s \operatorname{Li}_s(e^{-2\pi i x})={(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)\ge 0, 0 \le \textrm{Re}(x) < 1
and also for
\textrm{Re}(s)>1, \textrm{Im}(x)\le 0, 0 < \textrm{Re}(x) \le 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\over2i\pi}\right),

I've several comments/questions:

  • The restriction to 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)

[edit] 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 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 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\over2}\ +\ (-\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 Γ 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).

[edit] 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 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)

[edit] 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)

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