Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
113
questions with no upvoted or accepted answers
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Polylogarithm further generalized
Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
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Efficient calculation for Lerch Transcendent Expression
I've encountered:
$$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$
When trying to compute:
$$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
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Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function
$\def\W{\operatorname W} \def\Li{\operatorname{Li}} $
Interested by $\sum_\limits{n=1}^\infty\frac{\W(n^2)}{n^2}$, here is an example where Lagrange reversion applies to a Lambert W sum:
$$\W(x)=\ln(...
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Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$
It's possible to generalize these series?
$$\sum_{k=1}^{\infty}H_k^{(s)}x^k=\frac{\operatorname{Li}_s(x)}{1-x}$$
$$\sum_{k=1}^{\infty}H_k^2 x^k=\frac{\ln(1-x)^2+\operatorname{Li}_2(x)}{1-x}$$
Where:
$$...
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76
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Solving a set of implicit equations involving Polylogarithms
I have the following simultaneous equations:
\begin{aligned}
&H(\lambda) = a\, \text{Li}_{3/2}\left(b\frac{H(\lambda)}{F(\lambda)}\right), \; \\&H(\lambda) = c\, \text{Li}_{3/2}\left(d \, \...
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Dilogarithm Function on Negative Domain
I'm not that good with math, but somehow ended up solving for
$ \int { \ln { (\cosh x) } } \cdot dx $. This has led me to the answer described here. In my case, I need a solution for x > 1, ...
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Can $\text{Li}_2(x/y)$ be expressed as a sum of other $\text{Li}_2$?
I am looking at the following function:
$$f(x_1,x_2):=\text{Li}_2 \left( \frac{x_1}{x_2} \right). \tag{1}$$
I would like to know whether this function can be expressed as a sum of harmonic polylogs (...
- 697
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Closed-form for $\int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds$
In my partial answer to this question: Integral involving polylogarithm and an exponential, I arrive at the integral
$$ \int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds , ~~~~ (\ast) $$
where $a \in ...
- 753
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Difference of polylogarithms of complex conjugate arguments
I have the expression
$$\tag{1}
\operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*)
$$
Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
- 4,817
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Conjectured closed form for ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$
With Maple i find this closed form:
${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt {
2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2}
...
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Using Contour Integral to find the value of $\int_{-1}^{+1}\frac{\ln{(1+t)}}{t}dt$
$\newcommand{LogI}{\operatorname{Li}}$
We know that the value of $\LogI_{2}(-1)$ is -$\frac{\pi^2}{12}$ and $\LogI_{2}(1)$ is $\frac{\pi^2}{6}$. The value of the polylogarithms has already been ...
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Upper bounds regarding polylogarithm $Li_p(e^w)$ when $|w| > 2\pi$ and p negative real
$\textit{The Computations of Polylogarithms, 1992,Technical report UKC, University of Kent, Canterbury, UK}\\ $ by $\textbf{David C Wood}$ says that
$$
Li_p(e^w) = \sum_{n\geq 0} \zeta(p-n)\frac{w^n}{...
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Dilogarithm of a negative real number outside unit circle
The dilogarithm is defined in $\mathbb{C}$ as
$$
Li_2(z) = -\int_0^1 \frac{\ln(1 - zt)}{t} dt
$$
Because $1-zt \in \mathbb{C}$, then you can write $\ln(1 - zt) = \ln|1 - zt| + i·\arg(1 - zt)$
As ...
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Polylog identities
I'm looking for an algebraic identity (if it exists) that relates $\text{Li}_n(-z)$, $\text{Li}_n\left(\frac{1}{1+z}\right)$ and/or $\text{Li}_n\left(\frac{1}{z}\right)$ for $z > 0$ and $n \in {\...
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Different approach to compute $\int_0^1\frac{\ln(x)}{1+x}\text{Li}_2\left(\frac{1+x}2\right)\ dx$
The following integral $$I=\int_0^1\frac{\ln(x)}{1+x}\text{Li}_2\left(\frac{1+x}2\right)\ dx$$
was already evaluated by @Knas here where he found
$$I=-2\operatorname{Li}_4\left(\dfrac{1}{2}\right)-\...
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