All Questions
Tagged with polylogarithm special-functions
119
questions
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
2
votes
0
answers
55
views
Anti-derivative of a product of three logarithms and a rational function.
Let $a$ and $b$ be real numbers. The question is how do we compute the following anti-derivative:
\begin{equation}
{\mathfrak I}^{(a,b)}(x):=\int\frac{\log(x+a) \log(x+b)^2}{x} dx=?
\end{equation}
...
- 11.5k
1
vote
1
answer
75
views
Anti-derivative of a function that involves poly-logarithms.
Let $n\ge 1$ be an integer and let $0 < z < a$ be real numbers.
Let $Li_n(x):= \sum\limits_{l=1}^\infty z^l/l^n$ by the polylogarithm of order $n$.
The question is to find the following anti-...
- 11.5k
3
votes
3
answers
449
views
Indefinite integral $\int \arctan^2 x dx$ in terms of the dilogarithm function
I read about the integral
$$\int \arctan^2 x dx$$ in this old post: Evaluation of $\int (\arctan x)^2 dx$
By replacing
$$\arctan x = -\frac{i}{2}\left[\log(1+ix) - \log(i-ix)\right],$$
as suggested ...
- 7,613
1
vote
0
answers
77
views
An anti-derivative of a product of powers of two logarithms
Let $n\ge0$ and $m\ge0$ be integers. We intend to find the following integral:
\begin{equation}
{\mathfrak I}^{(m,n)}(z) := \int\limits_0^z [\log(1-\xi)]^m \cdot [\log(\xi)]^n d\xi
\end{equation}
Now,...
- 11.5k
2
votes
1
answer
229
views
Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha
I was thinking about the shape of integrals related with $\zeta(3)$ and Catalan's constant, I am saying those in section 3.1 of this Wikipedia. I was thinking in moments of higher order $x^k$ in the ...
user243301
18
votes
3
answers
930
views
Closed form for $\int_0^e\mathrm{Li}_2(\ln{x})\,dx$?
Inspired by this question and this answer, I decided to investigate the family of integrals
$$I(k)=\int_0^e\mathrm{Li}_k(\ln{x})\,dx,\tag{1}$$
where $\mathrm{Li}_k(z)$ represents the polylogarithm of ...
- 856
11
votes
6
answers
478
views
Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$
I stumbled upon the interesting definite integral
\begin{equation}
\int\limits_0^1 \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2
\end{equation}
Here is my proof of this result.
Let $u=\sin^{-1}(x)$ ...
- 2,428
5
votes
0
answers
161
views
Verifying closed form evaluation of an Ising-class multiple integral
For $n\in\mathbb{N}\land n\ge2$, define the so-called Ising-class integral of the third kind, $E_{n}$, via the sequence of $\left(n-1\right)$-dimensional integrals
$$E_{n}:=2\int_{\left[0,1\right]^{...
- 30.7k
3
votes
3
answers
1k
views
Proof of a dilogarithm identity
Through some experimentation, I've found:
$$\text{Li}_{2}(\sqrt{2}-1) \ + \text{Li}_{2}(1-\frac{1}{\sqrt{2}})\ =\frac{\pi^2}{8} - \frac{\ln^2(1 + \sqrt{2})}{2} - \frac{\ln^2{2}}{8} $$
I'm sure a ...
- 4,954
2
votes
0
answers
85
views
Natural proof of identity for $\text{Li}_2(x)+\text{Li}_2(1-x)$ [duplicate]
What's a natural way to compute $\text{Li}_2(x)+\text{Li}_2(1-x)$ in closed form ?
Once you know the answer $\text{Li}_2(x)+\text{Li}_2(1-x)=\frac{\pi^2}{6}-\log(x)\log(1-x)$ , computing the ...
- 35.9k
19
votes
3
answers
948
views
Proving that $\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$
I am trying to prove that
$$I=\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$$
where $\beta(s)$ is the Dirichlet Beta function and $G$ is the Catalan's constant. I managed ...
- 11.6k
5
votes
2
answers
438
views
Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$
Is there a closed form (free of polylogs and ...
- 369
5
votes
1
answer
753
views
On what domain is the dilogarithm analytic?
The series $\displaystyle\sum \dfrac{z^n}{n^2}$ converges for $\lvert z\rvert<1$ by the ratio test, meaning that the dilogarithm function $\text{Li}_2(z),$ which is equal to the series $\...
- 16.9k
8
votes
1
answer
461
views
A special value polylogarithm identity involving $\text{Li}_3(-1/2),\,\text{Li}_3(-1/3),\,\text{Li}_3(2/3),\,\text{Li}_2(-1/3),\,\text{Li}_2(2/3)$
I've found that
\begin{align}
\mathcal{L} = 2\operatorname{Li}_3\left(-\frac{1}{2}\right)+\operatorname{Li}_3\left(-\frac{1}{3}\right)+2\operatorname{Li}_3\left(\frac{2}{3}\right)+\operatorname{Li}_2\...
- 12.4k