All Questions
Tagged with polylogarithm special-functions
119
questions
3
votes
0
answers
184
views
How to simplify this polylog expression $\operatorname{Li}_4\left(\frac{z-1}{z}\right)$?
Evaluating this integral in Mathematica
$$i_2(z)=\int_0^z \frac{\log ^2(x) \log (1-x)}{1-x} \, dx\tag{1}$$
returns a mixture of polylogs up to order 4 and several log-terms.
In the region of ...
- 12.7k
1
vote
1
answer
347
views
Properties of Polylogarithm functions
I have the following expression,
$$\mathrm{Li}_3(-e^{+iy})-\mathrm{Li}_3(-e^{-iy})+\left[\mathrm{Li}_4(-e^{-iy})+\mathrm{Li}_4(-e^{+iy})\right],$$
where $0<y<\pi$. I know that this expression ...
- 514
4
votes
2
answers
258
views
How to prove $\int_0^\infty e^{-a q} \Gamma (0,q)^2 \, dq = -\frac{1}{a}\left(2 \operatorname{Li}_2(-a-1)+\frac{\pi ^2}{6}\right)$
In my study (https://math.stackexchange.com/a/3392284/198592) of pdfs for the harmonic mean of $n$ independent random variables $x_{i} \sim U(0,1)$ I discovered the lemma
$$f_{2}(a) = \int_0^{\infty }...
- 12.7k
7
votes
2
answers
228
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On the Clausen triple $8\rm{Cl}_2\left(\frac{\pi}2\right)+3\rm{Cl}_2\left(\frac{\pi}3\right)=12\,\rm{Cl}_2\left(\frac{\pi}6\right)$
While doing research on the Clausen function, I came across this nice identity,
$$8\operatorname{Cl}_2\left(\frac{\pi}2\right)+3\operatorname{Cl}_2\left(\frac{\pi}3\right)=12\operatorname{Cl}_2\left(\...
- 54.9k
6
votes
0
answers
182
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Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?
This was inspired by Oksana's post. Let, $$a = \ln 2 \quad\quad\\ b = \ln 3\quad\quad\\ c = \ln 5\quad\quad$$
then the following,
\begin{align}
A &= \text{Li}_3\left(\frac12\right)\\
B &= \...
- 54.9k
2
votes
1
answer
380
views
Expressions of G-BARNES
All people know expressions G-BARNES FUNCTION for example G(1/2), G(3/2) etc ... or G(1/4), G(3/4).
But someone know G(1/8), G(3/8), G(5/8) or G(7/8) in terms of Psi(1,1/8) ?
Thanks.
- 87
2
votes
0
answers
55
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Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?
The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
- 54.9k
7
votes
3
answers
615
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Closed-forms for the integral $\int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$?
(This is related to this question.)
Define the integral,
$$I_n = \int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$$
with polylogarithm $\operatorname{Li}_n(x)$. Given the Nielsen generalized polylogarithm ...
- 54.9k
12
votes
1
answer
611
views
More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$
In this post, the OP asks about the integral,
$$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$
I. User DavidH gave a beautiful (albeit long)...
- 54.9k
5
votes
4
answers
394
views
Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$
I am trying to evaluate $$P=\frac\pi2\sum_{n\geq1}\frac{{2n\choose n}}{4^n n^2}$$
I used the beta function to show that
$$P=\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$$
IBP:
$$P=\sin^{-1}(x)\...
- 17.3k
1
vote
0
answers
81
views
An anti-derivative involving an arc tangent, a square root and a rational function.
This question is similar to A generalized Ahmed's integral .
Let $a_1 \in {\mathbb R}_+$, $a_2 \in {\mathbb R}_+$ and $b_1 \in {\mathbb R}_+$. Consider the following integrals:
\begin{eqnarray}
{\...
- 11.5k
14
votes
1
answer
480
views
Yet another difficult logarithmic integral
This question is a follow-up to MSE#3142989.
Two seemingly innocent hypergeometric series ($\phantom{}_3 F_2$)
$$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{(-1)^n}{2n+1}\qquad \...
- 355k
13
votes
1
answer
461
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A logarithmic integral, generalization of a result of Shalev
As many of you are already aware, I and Marco Cantarini are currently working on the applications of fractional operators to hypergeometric series, extending the class of $\phantom{}_{p+1} F_p$s whose ...
- 355k
14
votes
1
answer
466
views
A peculiar Euler sum
I would like a hand in the computation of the following Euler sum
$$ S=\sum_{m,n\geq 0}\frac{(-1)^{m+n}}{(2m+1)(2n+1)^2(2m+2n+1)} \tag{1}$$
which arises from the computation of $\int_{0}^{1}\frac{\...
- 355k
4
votes
0
answers
74
views
Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?
Let us define a following function:
\begin{eqnarray}
{\mathcal J}^{(d)}(\vec{A}) := \int\limits_0^\infty e^{-u^2} \prod\limits_{\xi=1}^d erf(A_\xi u) du
\end{eqnarray}
for $\vec{A}:=\left(A_\xi\right)...
- 11.5k