All Questions
6
questions
1
vote
1
answer
164
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Integral involving product of dilogarithm and an exponential
I am interested in the integral
\begin{equation}
\int_0^1 \mathrm{Li}_2 (u) e^{-a^2 u} d u , ~~~~ (\ast)
\end{equation}
where $\mathrm{Li}_2$ is the dilogarithm. This integral arose in my attempt to ...
- 753
5
votes
1
answer
583
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Polylogarithms: How to prove the asympotic expression $ z \le \mathrm{Li}_{s}(z) \le z(1+2z 2^{-s}), \;z<-1, \;s \gg \log_2|z|$
For $|z| < 1, s > 0$ the polylogarithm has the power series
$$\mathrm{Li}_{s}(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots = z\left(1+ {z \over 2^s} + {...
- 18.9k
2
votes
0
answers
85
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Natural proof of identity for $\text{Li}_2(x)+\text{Li}_2(1-x)$ [duplicate]
What's a natural way to compute $\text{Li}_2(x)+\text{Li}_2(1-x)$ in closed form ?
Once you know the answer $\text{Li}_2(x)+\text{Li}_2(1-x)=\frac{\pi^2}{6}-\log(x)\log(1-x)$ , computing the ...
- 35.9k
4
votes
1
answer
330
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A possible dilogarithm identity?
I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before?
$$2 \left(\text{Li}_2\left(2-\sqrt{2}\right)+\text{...
- 44.4k
22
votes
4
answers
1k
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Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?
Ramanujan gave the following identities for the Dilogarithm function:
$$
\begin{align*}
\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) &=\frac{{...
- 11.6k
8
votes
2
answers
391
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Closed form of $\operatorname{Li}_2(\varphi)$ and $\operatorname{Li}_2(\varphi-1)$
I am trying to calculate the dilogarithm of the golden ratio and its conjugate $\Phi = \varphi-1$. Eg the solutions of the equation $u^2 - u = 1$.
From Wikipdia one has the following
\begin{align*}...
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