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Tagged with polylogarithm special-functions
119
questions
14
votes
6
answers
3k
views
Inverse of the polylogarithm
The polylogarithm can be defined using the power series
$$
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
$$
Contiguous polylogs have the ladder operators
$$
\operatorname{Li}_{s+1}(z) ...
- 12.5k
22
votes
4
answers
1k
views
Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?
Ramanujan gave the following identities for the Dilogarithm function:
$$
\begin{align*}
\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) &=\frac{{...
20
votes
7
answers
905
views
Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$
Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$
Numerically, it's about
$$\approx 111....
- 67.8k
0
votes
0
answers
39
views
Relations between Dilogarithms and Imaginary part of Hurwitz-Zeta function
I'm working through a paper that involves a problem concerning the calculation of the Imaginary part of the derivative of the Hurwitz-Zeta function $\zeta_H(z,a)$ with respect to $z$, evaluated at a ...
1
vote
1
answer
59
views
Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 4,249
2
votes
0
answers
68
views
Help verifying expression involving dilogarithms.
I need help verifying that the following equality holds:
$$Li_2(-2-2\sqrt2)+Li_2(3-2\sqrt2)+Li_2(\frac{1}{\sqrt2})-Li_2(-\frac{1}{\sqrt2})-Li_2(2-\sqrt2)-Li_2(-1-\sqrt2)-2Li_2(-3+2\sqrt2)$$
$$=$$
$$\...
- 359
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 ...
- 317
5
votes
1
answer
288
views
Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $
(This is related to this question).
How would one find the closed forms the integral
$$ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x?
$$
I tried using Nielsen Generalized Polylogarithm as mentioned ...
- 856
4
votes
0
answers
83
views
Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,702
5
votes
1
answer
193
views
Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
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 ...
- 25.8k
2
votes
0
answers
68
views
Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\int\frac{\log(x+a)}{x}\,dx$$
And wanted to make sure my solution is correct, and if not, where I went wrong in my process. Here is my work.
$$\int\frac{\log(x+a)}{x}\,...
- 1,123
2
votes
1
answer
229
views
Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha
I was thinking about the shape of integrals related with $\zeta(3)$ and Catalan's constant, I am saying those in section 3.1 of this Wikipedia. I was thinking in moments of higher order $x^k$ in the ...
- 30.7k
3
votes
1
answer
92
views
Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?
I just noticed that $B_{2n}(\frac12+ix)\in\mathbb R$, where:
$x\in\mathbb R$, $n\in\mathbb N$, and $B_n(x)$ is the $n$th Bernoulli Polynomial.
Why? Is there a simple, slick proof? Does it follow from ...
- 6,672
4
votes
0
answers
81
views
How to derive this polylogarithm identity (involving Bernoulli polynomials)?
How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
- 33.2k