All Questions
8 questions
0
votes
4
answers
237
views
Integrating $\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}} \,dx$
$$\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}}
\,dx$$
Now I have seen a similar integral here,
I am unable to deal with the $x^{\frac12}$ term in the denominator.
Here's one idea,
$$\int_0^1x^...
- 5,691
4
votes
2
answers
456
views
Harmonic sum with Dirichlet eta tail
The following problem is proposed by Cornel Valean:
$$\sum_{n=1}^{\infty} \frac{H_n}{2n+1}\left(\eta(2)- \overline{H}_n^{(2)}\right)$$
$$=2 G^2-2\ln(2) \pi G+\ln^2(2)\frac{\pi ^2}{6} +\frac{53}{1440}\...
- 26.6k
17
votes
3
answers
672
views
Evaluating $\int_0^{\frac{\pi}{2}}x^2 \cot x\ln(1-\sin x)\mathrm{d}x$
I was able to find
$$\int_0^{\frac{\pi}{2}}x^2 \cot x\ln(1-\sin x)\mathrm{d}x=-\frac14\sum_{n=1}^\infty\frac{4^n}{{2n\choose n}}\frac{H_{2n}}{n^3}$$
$$=5\operatorname{Li}_4\left(\frac12\right)-\frac{...
- 26.6k
3
votes
4
answers
226
views
How to prove this series identity $\sum_{n=1}^{\infty}\left(\frac{1}{n}\sum_{k=n+1}^{\infty}\frac{1}{k^3}\right)=\frac{\pi^{4}}{360}$?
I tried to prove this identity seemingly related to special functions which has been verified via Mathematica without a proof:
$$\sum_{n=1}^{\infty}\left(\frac{1}{n}\sum_{k=n+1}^{\infty}\frac{1}{k^3}\...
- 382
7
votes
4
answers
561
views
Prove series form of fractional harmonic numbers
Let $H_\alpha$ be the $\alpha$th fractional harmonic number so that
$$
H_\alpha = \int_0^1 \frac{1-x^\alpha}{1-x}\,\text dx.
$$
I want to directly show
$$
H_\alpha = \sum_{k=1}^\infty \frac{\alpha}{k(...
- 1,509
21
votes
1
answer
1k
views
Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$
Prove that:
$$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$
Using ...
- 5,617
25
votes
5
answers
1k
views
Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$
How can we prove that:
$$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
- 5,617
14
votes
2
answers
2k
views
Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $
Prove the following
$$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$
where
$$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
- 14.6k