All Questions
Tagged with polylogarithm special-functions
119
questions
4
votes
0
answers
220
views
How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?
I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination
$$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
- 2,017
1
vote
0
answers
117
views
Closed-form for $\int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds$
In my partial answer to this question: Integral involving polylogarithm and an exponential, I arrive at the integral
$$ \int_0^{a^2} \mathrm{Ei} (-s) \frac{1 - e^s}{s} ds , ~~~~ (\ast) $$
where $a \in ...
- 753
1
vote
1
answer
164
views
Integral involving product of dilogarithm and an exponential
I am interested in the integral
\begin{equation}
\int_0^1 \mathrm{Li}_2 (u) e^{-a^2 u} d u , ~~~~ (\ast)
\end{equation}
where $\mathrm{Li}_2$ is the dilogarithm. This integral arose in my attempt to ...
- 753
1
vote
0
answers
59
views
Difference of polylogarithms of complex conjugate arguments
I have the expression
$$\tag{1}
\operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*)
$$
Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
- 4,817
4
votes
2
answers
180
views
Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$
I managed here to prove $$\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$$
but what I did was converting the LHS integral to a ...
- 25.8k
5
votes
1
answer
248
views
Closed form evaluation of a trigonometric integral in terms of polylogarithms
Define the function $\mathcal{K}:\mathbb{R}\times\mathbb{R}\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\rightarrow\mathbb{R}$ via the definite ...
- 30.7k
2
votes
1
answer
247
views
Finding a closed-form for the sum $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}$
Let $\mathcal{S}$ denote the sum of the following alternating series:
$$\mathcal{S}:=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}\approx-1.392562725547,$$
where $H_{n}$ denotes the $n$-...
- 30.7k
3
votes
1
answer
502
views
Generating function of the polylogarithm.
Let $\operatorname{Li}_s(z)$ denote the polylogarithm function
$$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}.$$
Does there exists a closed form or a known function which generates the ...
- 812
3
votes
2
answers
311
views
How can I evaluate $\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x$
I've been trying to find and prove that:
$$\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x=\pi \operatorname{Li}_2\left(\frac{1-\sqrt{2}}{2}\right)-\frac{\pi }{2}\left(\...
- 358
2
votes
1
answer
126
views
Closed form evaluation of a class of inverse hyperbolic integrals
Define the function $\mathcal{I}:\mathbb{R}_{>0}^{2}\rightarrow\mathbb{R}$ via the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\...
- 30.7k
4
votes
3
answers
257
views
Arctan integral $ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$
Is there a closed form for the integral
$$ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$$
for $\forall k \ge 1 $?
Well, I was able to get the closed form for the case where $|k|\le1$, and it is of ...
- 247
3
votes
1
answer
368
views
Challenging integral $I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx$
My friend offered to solve this integral.
$$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
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
2
answers
83
views
Closed-form expressions for the zeros of $\text{Li}_{-n}(x)$?
Consider the first few polylogarithm functions $\text{Li}_{-n}(x)$, where $-n$ is a negative integer and $x\in\mathbb R$ (plotted below). Observation suggests that $\text{Li}_{-1}(x)$ has one zero (at ...
- 6,672
2
votes
0
answers
154
views
Evaluating a variant of the polylogarithm
Consider the infinite sum :
$$\sum_{m=1}^{\infty}\binom{m}{my}\frac{z^{m}}{m^{s}}\;\;\;\;s\in\mathbb{C},\;\;\;\;|z|<\frac{1}{2},\;\;\;\;\;0<y<1$$
I want to evaluate this summation in terms ...
