All Questions
Tagged with polylogarithm integration
271
questions
17
votes
1
answer
1k
views
A rare integral involving $\operatorname{Li}_2$
A rare but interesting integral problem:
$$\int_{0}^{1}
\frac{\operatorname{Li}_2(-x)-
\operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x)
-\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} }
=\...
- 5,370
8
votes
2
answers
494
views
Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$
Prove the integral
$$\int_{1}^{\infty} \frac{1}{1+x^2}
\frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{
\pi^2+\ln^2\left ( \frac{x-1}{2} \right ) }\text{d}x
=\frac{96C\ln2+7\pi^3}{12(\pi^2+...
- 5,370
9
votes
1
answer
329
views
Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$
The following question:
How to compute the harmonic series $$\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$$
where $H_n=\sum_{k=1}^n\frac{1}{k}$ and $H_n^{(2)}=\sum_{k=1}^n\frac{1}{k^2}$, was ...
- 25.8k
6
votes
1
answer
494
views
Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?
We know how hard these integrals
$$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx;
...$$
can be. So I decided to come up with a ...
- 25.8k
8
votes
2
answers
428
views
Evaluating $\int_0^\infty\frac{\tan^{-1}av\cot^{-1}av}{1+v^2}\,dv$
The Weierstrass substitution stuck in my head after I used it to prove the rigidity of the braced hendecagon (and tridecagon). Thus I had another look at this question which I eventually answered in a ...
- 104k
4
votes
2
answers
180
views
Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$
I managed here to prove $$\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$$
but what I did was converting the LHS integral to a ...
- 25.8k
6
votes
0
answers
362
views
Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 5,370
12
votes
3
answers
460
views
How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$
I am trying evaluating this
$$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\ \text{d}x.$$
For $k=1$, there has
$$J(1)=\frac{\pi^4}{96}.$$
Maybe $J(k)$ ...
- 5,370
3
votes
0
answers
291
views
Evaluate $\int_{1}^{\infty}\frac{\operatorname{Li}_3(-x)\ln(x-1)}{1+x^2}\text{d}x$
Using $$
\operatorname{Li}_3(-x)
=-\frac{x}{2}\int_{0}^{1}\frac{\ln^2t}{1+tx}
\text{d}t
$$
It might be
$$
-\frac{1}{2}\int_{0}^{1}\ln^2t
\int_{1}^{\infty}\frac{x\ln(x-1)}{(1+tx)(1+x^2)}\text{d}x\text{...
- 5,370
7
votes
2
answers
338
views
evaluate $\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$
I came across this integral:
$$\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx=\frac{1}{2}\int_{-1}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$$
One way to evaluate is to start with the ...
- 528
8
votes
1
answer
422
views
Integral $\int _0^1\frac{\ln \left(x\right)\text{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x$
A comrade sent me this conjecture
$$\int _0^1\frac{\ln \left(x\right)\operatorname{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x=3\zeta (2)\left(4\ln \left(A\right)-1\...
- 241
3
votes
2
answers
311
views
How can I evaluate $\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x$
I've been trying to find and prove that:
$$\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x=\pi \operatorname{Li}_2\left(\frac{1-\sqrt{2}}{2}\right)-\frac{\pi }{2}\left(\...
- 358
2
votes
1
answer
126
views
Closed form evaluation of a class of inverse hyperbolic integrals
Define the function $\mathcal{I}:\mathbb{R}_{>0}^{2}\rightarrow\mathbb{R}$ via the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\...
- 30.7k
4
votes
3
answers
257
views
Arctan integral $ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$
Is there a closed form for the integral
$$ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$$
for $\forall k \ge 1 $?
Well, I was able to get the closed form for the case where $|k|\le1$, and it is of ...
- 247
7
votes
1
answer
196
views
Iterated integral involving polylogarithms
To establish notation the polylogarithm Li$_n(x)$ has the power series expansion
$$ \text{Li}_n(x)= \sum_{k=1}^\infty \frac{x^k}{k^n} $$
and the Riemann zeta can be considered the special value $\zeta(...
- 6,760