Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
468
questions
3
votes
0
answers
38
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 ...
- 5,328
4
votes
0
answers
118
views
Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$
How to show that
$$\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx=\frac{5\pi^3}{64}+\frac{\pi}{16}\ln^2(2)-4\,\text{G}\ln(2)$$
without breaking up the integrand since we already know:
$$\int_0^1\...
- 22k
2
votes
2
answers
235
views
Find close form for $\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$
I am trying to find a closed form for the integral,$$\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$$
I tried using the sub, $x=\frac{1-x}{1+x}$ but it was to no avail. I also tried the trig sub, $x=\tan(...
- 121
11
votes
0
answers
337
views
Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?
Using
$$\text{Li}_{2a+1}(x)-\text{Li}_{2a+1}(1/x)=\frac{i\,\pi\ln^{2a}(x)}{(2a)!}+2\sum_{k=0}^a \frac{\zeta(2a-2k)}{(2k+1)!}\ln^{2k+1}(x)\tag{1}$$
and
$$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
- 22k
1
vote
1
answer
29
views
Proving $-\text{Li}_2(x^{-1}-1)+\text{Li}_2(1-x^{-1})-\text{Li}_2(2 x)+\text{Li}_2(2-2 x)+\text{Li}_2(2) = i \pi \log(x)$ for $x>1/2$.
While working on a physics problem, I have stumbled upon the following identity. For $x>\frac{1}{2}$ note
$$
-\text{Li}_2\left(\frac{1}{x}-1\right)+\text{Li}_2\left(1-\frac{1}{x}\right)-\text{Li}_2(...
- 2,580
3
votes
0
answers
125
views
How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?
I have been working on a problem in number theory that I have reduced to the problem of showing that the two-term linear combination
$$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
- 1,829
0
votes
1
answer
42
views
How is $(2^a)^{\lg n} = n^a$? [closed]
I was learning from Introduction to Algorithms (Chapter 3 under the topic “Logarithms”) and came across this expression.
$$
\lim_{ n \to 0 }{\frac{\lg^b n}{(2^a)^{\lg n}}}
=
\lim_{n \to 0} \frac{...
- 315
0
votes
0
answers
30
views
Do Polylogarithms Always Converge
I have been reading the Wikipedia page dedicated to polylogarithms to understand the following paper. I am trying to understand the first term in equation (6) (reproduced below):
$$
P(\lambda) = \...
- 720
6
votes
2
answers
109
views
Series expansion of $\text{Li}_3(1-x)$ at $x \sim 0$
My question is simple, but maybe hard to answer. I would like to have a series expansion for $\text{Li}_3 (1-x)$ at $x \sim 0$ in the following form:
$$\text{Li}_3 (1-x) = \sum_{n=0} c_n x^n + \log x \...
- 667
0
votes
1
answer
64
views
Closed form for $\rm{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)$
In my personal research with Maple i find this closed form :
$$\operatorname{Li }_2\left( -{\frac {i\sqrt {3}}{3}} \right)={\frac {{\pi}^{2}}{24}}+{\frac {\ln \left( 2 \right) \ln \left( 3
\right) }...
- 193
1
vote
0
answers
18
views
Canonical reference for algebraic theory of polylogs?
I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to ...
- 37.5k
1
vote
2
answers
115
views
Evaluating improper integral $\int_0^1 \frac{\log(x)}{x+\alpha}\; dx$ for small positive $\alpha$
Let $\alpha$ be a small positive real number.
How do I obtain
$$ I = \int_0^1 \frac{\log(x)}{x+\alpha}\; dx = -\frac{1}{2}(\log\alpha)^2 - \frac{\pi^2}{6} - \operatorname{Li}_2(-\alpha)$$? Maxima told ...
- 23
18
votes
0
answers
683
views
Are these generalizations known in the literature?
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
- 22k
0
votes
1
answer
127
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Evaluate: ${{\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx}}$
Evaluate:
$${{I=\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx.}}$$
The answer is given below:
$$
I=-\frac{7}{12}\pi^4\ln^2(2)-\...
- 1,858
1
vote
1
answer
150
views
Can this formula for $\zeta(3)$ be proven or simplified further?
This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
- 4,688
9
votes
2
answers
262
views
Proving $\int_0^{1/2}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3)$
By comparing some results, I found that
$$\int_0^{\frac12}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3).\tag{1}$$
I tried to prove it starting with applying IBP:...
- 22k
1
vote
0
answers
102
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 ...
- 619
1
vote
1
answer
94
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 ...
- 619
3
votes
1
answer
28
views
Sum of Fermi-Dirac integrals with opposite chemical potentials: closed form (Le Bellac eq. 1.13)
I am trying to reproduce the result of eq. (1.13) in Le Bellac's Thermal Field Theory book to compute the grand canonical potential of a gas of massless fermions:
$$
Ω = - \frac{V T^4}{6 π^2} \int_0^\...
- 31
12
votes
1
answer
469
views
General expressions for $\mathcal{L}(n)=\int_{0}^{\infty}\operatorname{Ci}(x)^n\text{d}x$
Define $$\operatorname{Ci}(x)=-\int_{x}^{
\infty} \frac{\cos(y)}{y}\text{d}y.$$
It is easy to show
$$
\mathcal{L}(1)=\int_{0}^{\infty}\operatorname{Ci}(x)\text{d}x=0
$$
and
$$\mathcal{L}(2)=\int_{0}^{\...
- 1,858
6
votes
0
answers
144
views
A generalized "Rare" integral involving $\operatorname{Li}_3$
In my previous post, it can be shown that
$$\int_{0}^{1}
\frac{\operatorname{Li}_2(-x)-
\operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x)
-\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} }
=\...
- 1,858
11
votes
1
answer
490
views
A rare integral involving $\operatorname{Li}_2$
A rare but interesting integral problem:
$$\int_{0}^{1}
\frac{\operatorname{Li}_2(-x)-
\operatorname{Li}_2(1-x)+\ln(x)\ln(1+x)+\pi x\ln(1+x)
-\pi x\ln(x)}{1+x^2}\frac{\text{d}x}{\sqrt{1-x^2} }
=\...
- 1,858
7
votes
2
answers
322
views
Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$
Prove the integral
$$\int_{1}^{\infty} \frac{1}{1+x^2}
\frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{
\pi^2+\ln^2\left ( \frac{x-1}{2} \right ) }\text{d}x
=\frac{96C\ln2+7\pi^3}{12(\pi^2+...
- 1,858
1
vote
0
answers
45
views
Difference of polylogarithms of complex conjugate arguments
I have the expression
$$\tag{1}
\operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*)
$$
Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
- 3,368
3
votes
1
answer
70
views
Identity involving PolyLog functions
The following is a common identity between dilogarithms:
$$
\text{Li}_2(y)+\text{Li}_2\left(\frac{y}{y-1}\right)+\frac{1}{2} \log ^2(1-y)=0\quad \text{with}\quad 0<y<1\,.
$$
Similarly, with some ...
- 351
1
vote
0
answers
26
views
Asymptotics of Geometric Distribution Moments
It is known (e.g., see Wikipedia) that the $k$th moment of a Geometric distribution with success probability $p$ is
$$\mathbb{E}\left[X^k\right] = \sum_{j = 0}^\infty j^k \cdot p(1-p)^j = p \cdot\text{...
- 6,218
9
votes
1
answer
258
views
Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$
The following question:
How to compute the harmonic series $$\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$$
where $H_n=\sum_{k=1}^n\frac{1}{k}$ and $H_n^{(2)}=\sum_{k=1}^n\frac{1}{k^2}$, was ...
- 22k
0
votes
0
answers
33
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 ...
- 1
0
votes
1
answer
59
views
How to solve a Non-algebraic equation?
While working with exponential growth and decay, I encountered a problem where I need to solve an equation involving logarithm. I could not separate or could not make it explicit.
$y=\frac{\ln((1+r)(1-...
- 168
0
votes
0
answers
130
views
When are $\sqrt{-\log x}^{1+\varepsilon}\operatorname{Li}_{1/2}(x)$ and $\sqrt{-\log x}\operatorname{Li}_\nu(x)$ strictly increasing?
[Cross-posted on MathOverflow]
This problem (posed by a Discord user) originates from the study of the fire-diffuse-fire model of calcium dynamics. Briefly, I was told that the goal was to prove there ...
- 25.4k
6
votes
1
answer
408
views
Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?
We know how hard these integrals
$$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx;
...$$
can be. So I decided to come up with a ...
- 22k
1
vote
0
answers
90
views
Conjectured closed form for ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$
With Maple i find this closed form:
${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt {
2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2}
...
- 193
0
votes
1
answer
104
views
Evaluate $\int_{{\frac {\pi}{8}}}^{{\frac {7\,\pi}{8}}}\!{\frac {\ln \left( 1- \cos \left( t \right) \right) }{\sin \left( t \right) }}\,{\rm d}t$
I'm interested in this integral: $\int_{{\frac {\pi}{8}}}^{{\frac {7\,\pi}{8}}}\!{\frac {\ln \left( 1- \cos \left( t \right) \right) }{\sin \left( t \right) }}\,{\rm d}t$
I found this particular ...
- 193
8
votes
1
answer
309
views
Evaluating $\int_0^1\frac{\operatorname{Li}_2(x)\ln(1+x)}x\,dx$
Well, I've been trying to solve the following integral:
\begin{equation*}
\int_0^1\frac{\text{Li}_3(x)}{1+x}\mathrm dx,
\end{equation*}
where by integration by parts, making $u=\text{Li}_3(x)$ and $\...
- 461
5
votes
2
answers
263
views
Evaluating $\int_0^\infty\frac{\tan^{-1}av\cot^{-1}av}{1+v^2}\,dv$
The Weierstrass substitution stuck in my head after I used it to prove the rigidity of the braced hendecagon (and tridecagon). Thus I had another look at this question which I eventually answered in a ...
- 89.9k
4
votes
2
answers
142
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 ...
- 22k
4
votes
1
answer
100
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 ...
- 27.1k
6
votes
0
answers
315
views
Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 1,858
12
votes
4
answers
416
views
How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$
I am trying evaluating this
$$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\ \text{d}x.$$
For $k=1$, there has
$$J(1)=\frac{\pi^4}{96}.$$
Maybe $J(k)$ ...
- 1,858
3
votes
0
answers
269
views
Evaluate $\int_{1}^{\infty}\frac{\operatorname{Li}_3(-x)\ln(x-1)}{1+x^2}\text{d}x$
Using $$
\operatorname{Li}_3(-x)
=-\frac{x}{2}\int_{0}^{1}\frac{\ln^2t}{1+tx}
\text{d}t
$$
It might be
$$
-\frac{1}{2}\int_{0}^{1}\ln^2t
\int_{1}^{\infty}\frac{x\ln(x-1)}{(1+tx)(1+x^2)}\text{d}x\text{...
- 1,858
6
votes
2
answers
311
views
evaluate $\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$
I came across this integral:
$$\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx=\frac{1}{2}\int_{-1}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$$
One way to evaluate is to start with the ...
- 510
8
votes
1
answer
349
views
Integral $\int _0^1\frac{\ln \left(x\right)\text{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x$
A comrade sent me this conjecture
$$\int _0^1\frac{\ln \left(x\right)\operatorname{Li}_2\left(x\right)}{x\left(4\pi ^2+\ln ^2\left(x\right)\right)}\:\mathrm{d}x=3\zeta (2)\left(4\ln \left(A\right)-1\...
- 241
2
votes
1
answer
151
views
Finding a closed-form for the sum $\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}$
Let $\mathcal{S}$ denote the sum of the following alternating series:
$$\mathcal{S}:=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}H_{2n}}{n^{4}}\approx-1.392562725547,$$
where $H_{n}$ denotes the $n$-...
- 27.1k
3
votes
1
answer
202
views
Generating function of the polylogarithm.
Let $\operatorname{Li}_s(z)$ denote the polylogarithm function
$$\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}.$$
Does there exists a closed form or a known function which generates the ...
- 710
2
votes
2
answers
184
views
How can I evaluate $\int _0^1\frac{\operatorname{Li}_2\left(-x^2\right)}{\sqrt{1-x^2}}\:\mathrm{d}x$
I've been trying to find and prove that:
$$\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(\...
- 250
2
votes
1
answer
76
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
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\...
- 27.1k
4
votes
3
answers
179
views
Arctan integral $ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$
Is there a closed form for the integral
$$ \int_{0}^{\infty}\frac{\arctan(x)}{x^{2}+k^{2}}$$
for $\forall k \ge 1 $?
Well, I was able to get the closed form for the case where $|k|\le1$, and it is of ...
- 119
7
votes
1
answer
163
views
Iterated integral involving polylogarithms
To establish notation the polylogarithm Li$_n(x)$ has the power series expansion
$$ \text{Li}_n(x)= \sum_{k=1}^\infty \frac{x^k}{k^n} $$
and the Riemann zeta can be considered the special value $\zeta(...
- 6,503
0
votes
1
answer
52
views
The rate of convergence of the remainder of the power series for the Polylog function [closed]
Let $0<p<1$ be a positive real number strictly smaller than one and $q>0$ be a positive real number. Consider the series
$$
\mathsf{Li}_{-q}(p) = \sum_{\ell=1}^{+\infty}\ell^{q}p^{\ell}
$$
...
- 1,044
9
votes
1
answer
182
views
Is there a closed-form for $\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}$?
So far I've got this:
$$\sum _{k=1}^{\infty }\frac{\operatorname{Si}\left(k\right)}{k^2}=\int _0^1\left(\sum _{k=1}^{\infty }\frac{\sin \left(kx\right)}{k^2}\right)\frac{1}{x}\:dx$$
$$=\int _0^1\frac{\...
- 250