All Questions
Tagged with polylogarithm closed-form
125
questions
20
votes
4
answers
1k
views
Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$
How can I prove that
$$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$
I think this post can help me, but I'm not sure.
- 5,626
23
votes
1
answer
533
views
Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$
Let
$$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$
where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as
$$\begin{...
- 13.2k
10
votes
1
answer
600
views
Indefinite integral $\int \arcsin \left(k\sin x\right) dx$
It would take too long to explain the context reasonably well - but in short, this integral, or rather its equivalent
$$\int\frac{x\cos x\,dx}{\sqrt{1-k^2\sin^2x}},\qquad 0<k<1$$
is related to ...
- 53.4k
8
votes
7
answers
907
views
Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$
How would you solve the following
$$\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$$
I might be able to relate the integral to Euler sums .
- 14.4k
9
votes
6
answers
702
views
Computing the value of $\operatorname{Li}_{3}\left(\frac{1}{2} \right) $
How to prove the following identity
$$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)\,?$$
Where ...
- 6,050