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22
votes
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answer
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A multiple integral that seems related to the $\zeta$ function at even integers
I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
- 545
5
votes
3
answers
845
views
Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
I uploaded this question here and here from my old account.
Let $\psi^{(1)}$ be the trigamma function defined by
\begin{equation}
\tag{1}
\psi_1(z) = -\int\limits_0^1 \frac{x^{z-1} \ln x}{1-x} \, dx.
\...
- 224
0
votes
0
answers
76
views
Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$
How could one calculate the closed form solution of this integral:
$\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$
Here the integral is ...
- 71