All Questions
Tagged with polylogarithm integration
271
questions
8
votes
1
answer
245
views
Is there a closed-form for $\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}$?
So far I've got this:
$$\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}=\int _0^1\left(\sum _{k=1}^{\infty }\frac{\sin \left(kx\right)}{k^2}\right)\frac{1}{x}\:dx$$
$$=\int _0^1\frac{\...
- 358
0
votes
0
answers
84
views
A Logarithmic Definite Integral Problem
I was trying to solve the integral
$$
A=\int_0^{\infty}\frac{x}{e^x + 1}\, dx.
$$
Firstly, I was trying to solve the indefinite one
$$
I(x)=\int\frac{x}{e^x + 1}\, dx.
$$
By subtitution $t = e^x$, ...
- 113
4
votes
3
answers
211
views
Does $\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\right)\,\ln\left(1-\frac{\cos^2\phi}{C^2}\right)$ have a closed form?
I am wondering if anyone has a nice way of approaching the following definite integral $\newcommand{\dilog}{\operatorname{Li}_2}$
$$\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\...
- 179
3
votes
3
answers
392
views
Logarithmic integral $ \int_0^1 \frac{x\ln x\ln(1+x)}{1+x^2}\ \mathrm{d}x $
I found this integral weeks ago.
$$ \int_0^1 \dfrac{x\ln(x)\ln(1+x)}{1+x^2}\ \mathrm{d}x $$
I tried to solve this integral using various series representation and ended up with a complicated double ...
- 2,149
7
votes
3
answers
280
views
Compute the double sum $\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)$
I am trying to compute the following double sum
$$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$
I proceeded as following
$$\sum_{n=1}^{\infty}\sum_{m=1}^...
- 2,811
3
votes
1
answer
368
views
Challenging integral $I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx$
My friend offered to solve this integral.
$$I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx=\frac{\pi^4}{32}-{4G^2} $$
Where G is the Catalan's constant.
$$I=\int _0^{\infty }\frac{\arctan ^2\left(u\...
- 5,507
3
votes
4
answers
475
views
How to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}H_{2n}$
Edit
In this post I computed the following integral
$$\int_{0}^{1}\frac{\log(1-x)\log(1-x^2)}{x}dx=\frac{11}{8}\zeta(3)$$
Now I am trying to compute
$$\boxed{\int_{0}^{1}\frac{\log(1-x)\log(1-x^4)}{x}...
- 2,811
0
votes
1
answer
177
views
$\int \frac{\ln(1-\sin(2x))}{\sin(2x)}\, dx$
$$\int \frac{\ln(1-\sin(2x))}{\sin(2x)}\,\mathrm{d}x$$
I tried to find the solution through WolframAlpha and I got the result, but I do not know how the solution is:
$$1/2 (2 \operatorname{Li}_2(1 - \...
- 59
6
votes
2
answers
322
views
Derive $\int_0^1 \frac{\ln(\sqrt2-1)-(\sqrt2-x)\ln x}{(\sqrt2-x)^2-1}\,dx=\frac{\pi^2}6+\frac14\ln^2(\sqrt2-1) $
I obtained the integral
$$\int_0^1 \frac{\ln(\sqrt2-1)-\ln(x)(\sqrt2-x)}{(\sqrt2-x)^2-1}\,dx=\frac{\pi^2}6+\frac14\ln^2(\sqrt2-1)
$$
as a by-product while carrying out some complex analysis on an ...
- 99.7k
-1
votes
3
answers
144
views
Stuck with integral involving Polylogarithms $\int_{0}^{\infty}\frac{2t}{e^{t\pi}+1} \,dt$
For a research work I ended up needing to give a proof of Zeta's trivial zeros, in order to do so I tried using the Abel-Plana formula.
$\zeta(s)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty} \frac{\sin(...
- 37
4
votes
1
answer
286
views
Evaluate $\int^1_0 x^a (1-x)^b \operatorname{Li}_2 (x)\, \mathrm dx$
For what $a,b$ the integral
$$\int^1_0 x^a(1-x)^b\operatorname{Li}_2 (x)\, \mathrm dx$$
has a closed form solution? I tried to solve it by expanding dilogarithm function, or by reducing it to linear ...
- 2,903
1
vote
1
answer
194
views
Definite integral for $\zeta(3)$
By making use of Mathematica, I detected the following integral expression for zeta(3):
$$\int_0^1\frac{\log(x)}{1+x}\log\left(\frac{2+x}{1+x}\right)dx=-\frac5{12}\zeta(3).$$
Any proof of it would be ...
1
vote
1
answer
83
views
Integral of a modified softplus function
In a manuscript I am currently reading, the authors propose a modified softplus function
$$g(a)=\frac{\log\left(2^a +1 \right)}{\log(2)}$$
for some $a \in \mathbb{R}$. The authors then claim that if $...
- 961
4
votes
0
answers
341
views
How to evaluate $\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$
I am trying to evaluate
$$\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$$
But I am not sure what to do since integration by parts is not possible here.
I tried using a ...
user809806
5
votes
1
answer
223
views
How to evaluate $\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\right)\:dx$
I want to evaluate $$\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\right)\:dx$$
But I've not been successful in doing so, what I tried is
$$\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\...
user809806