All Questions
4
questions
17
votes
2
answers
894
views
A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ...
Question: How to show the relation
$$
J:=\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt
=-\frac 1{64}\pi^4
$$
(using a "minimal industry" of relations, ...
- 34.2k
2
votes
2
answers
226
views
Evaluation of a log-trig integral in terms of the Clausen function (or other functions related to the dilogarithm)
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ via the definite integral
$$\mathcal{I}{\left(a,\theta\right)}:=\int_{0}^{\theta}\mathrm{d}\varphi\,\ln{\left(1-2a\cos{\left(\...
- 30.7k
2
votes
1
answer
111
views
Evaluate $\sum_{n\geq1} \frac{(-1)^{n+1}H_n^2}{(n+1)^2}$.
I am looking for a closed for $$\sum_{n\geq1} \frac{(-1)^{n+1}H_n^2}{(n+1)^2}.$$ I believe there is a closed form for the sum as we have seen in [1] which poses as, presumably, a more difficult sum of ...
- 125
2
votes
1
answer
686
views
Calculate a $\operatorname{Li}_{2}(-1)$ using Integral Representation
$\newcommand{LogI}{\operatorname{Li}}$
I know that $\LogI_{2}(-1)=-\frac{\pi^2}{12}$, but I have never seen a proof of this result without using a functional identity of the Dilogarithm or a series ...
- 436