Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
203
questions
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A group of important generating functions involving harmonic number.
How to prove the following identities:
$$\small{\sum_{n=1}^\infty\frac{H_{n}}{n^2}x^{n}=\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)+\ln(1-x)\operatorname{Li}_2(1-x)+\frac12\ln x\ln^2(1-x)+\zeta(3)...
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42
votes
7
answers
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Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$
$\qquad\qquad$ Is there any closed form expression for the imaginary part of $~\text{Li}_3\bigg(\dfrac{1+i}2\bigg)$ ?
Motivation: We already know that $~\Re\bigg[\text{Li}_3\bigg(\dfrac{1+i}2\bigg)\...
- 48.5k
44
votes
2
answers
3k
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Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$
We have the following result ($\text{Li}_{n}$ being the polylogarithm):
$$\tag{*}\small{ \int_0^1 \log^2 (1-x) \log^2 x \log^3(1+x) \frac{dx}{x} = -168 \text{Li}_5(\frac{1}{2}) \zeta (3)+96 \text{Li}...
- 19.1k
4
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3
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Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$
How to prove
$$\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$$
I havent started, any hints?
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14
votes
2
answers
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Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $
Prove the following
$$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$
where
$$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
- 14.4k
49
votes
9
answers
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Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $
I'm looking for a closed form of this integral.
$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$
where $\operatorname{Li}_2$ is the dilogarithm function.
A numerical ...
- 12.4k
31
votes
3
answers
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What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?
Some time ago I asked How to find $\displaystyle{\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$.
Thanks to great effort of several MSE users, we now know that
\begin{align}
\int_0^1\frac{\ln^3(1+x)\,\ln ...
- 5,342
11
votes
2
answers
605
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Compute $\int_0^{\pi/2} x^2\left(\sum_{n=1}^\infty (-1)^{n-1} \cos^n(x)\cos(nx)\right)dx$
How to prove
$$I=\int_0^{\pi/2} x^2\left(\sum_{n=1}^\infty (-1)^{n-1} \cos^n(x)\cos(nx)\right)dx=\frac16\left(\frac{\pi^3}{12}-\pi\operatorname{Li}_2\left(\frac13\right)\right)$$
This problem is ...
- 25.8k
6
votes
3
answers
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Short calculation of the dilogarithm?
Is there a nice way to implement the dilogarithm function for real values, without actually performing the integration?
A series solution would have been nice, but the series around $0$ has a ...
- 33.1k
43
votes
7
answers
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Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$
In the following thread
I arrived at the following result
$$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$
Defining
$$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, H_k^{(1)}\...
- 14.4k
17
votes
3
answers
825
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A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$
In evaluating the integral given here it would seem that:
$$\operatorname{Re} \operatorname{Li}_4 (1 + i) \stackrel{?}{=} -\frac{5}{16} \operatorname{Li}_4 \left (\frac{1}{2} \right ) + \frac{97}{...
- 11.8k
12
votes
2
answers
450
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Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?
(This summarizes my posts on Nielsen polylogs.)
I. Question 1: How to complete the table below? Consider the special cases $z=-1$ and $z=\frac12$. Given the Nielsen generalized polylogarithm,
$$S_{n,...
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20
votes
3
answers
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Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$
I numerically discovered the following conjecture:
$$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
- 47.3k
17
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8
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About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$
I came across the following Integral and have been completely stumped by it.
$$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$
I'm extremely sorry, but the only thing I noticed was that the ...
14
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4
answers
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Compute $\int_0^{1/2}\frac{\left(\operatorname{Li}_2(x)\right)^2}{x}dx$ or $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^32^n}$
Prove that
I encountered this integral while working on the sum $\displaystyle \sum_{n=1}^\infty \frac{H_n^{(2)}}{n^32^n}$. Both of the integral and the sum were proposed by Cornel Valean:
The ...
- 25.8k
10
votes
2
answers
670
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A surprising dilogarithm integral identity arising from a generalised point enclosure problem
This question asked:
What is the probability that three points selected uniformly randomly on the unit circle contain a fixed point at distance $x$ from the circle's centre?
I answered that ...
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8
votes
0
answers
413
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More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$
I. In this post, the OP asks about the particular log sine integral,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
6
votes
1
answer
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On generalizing the harmonic sum $\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n = S_{k-1,2}(1)+\zeta(k+1)$ when $z=1$?
Given the nth harmonic number $ H_n = \sum_{j=1}^{n} \frac{1}{j}$. In this post it asks for the evaluation,
$$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\tfrac{5}{4}\zeta(4)$$
while this post and this ...
- 54.9k
41
votes
2
answers
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Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$
Here is another integral I'm trying to evaluate:
$$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$
A numeric approximation is:
$$I\approx-0....
- 47.3k
32
votes
1
answer
818
views
On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$
Motivation
$\newcommand{Li}{\operatorname{Li}}$
It is already known that:
$$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$
$$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$
And by this question, ...
- 8,679
25
votes
2
answers
729
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Definite integral of arcsine over square-root of quadratic
For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{a}\frac{\arcsin{\left(2x-1\right)}\,\mathrm{d}x}{\...
- 30.7k
22
votes
2
answers
3k
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Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$$\operatorname{Re}\big[\operatorname{Li}_2\left(i\right)\big]=-\frac{\pi^2}{48},\hspace{1em}\...
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22
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4
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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:
$$
\begin{align*}
\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) &=\frac{{...
- 11.6k
20
votes
4
answers
1k
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Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$
How can I prove that
$$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$
I think this post can help me, but I'm not sure.
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17
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2
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894
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A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ...
Question: How to show the relation
$$
J:=\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt
=-\frac 1{64}\pi^4
$$
(using a "minimal industry" of relations, ...
- 34.2k
8
votes
2
answers
627
views
Closed form of a series (dilogarithm)
We are all aware of the dilogarithm function (Spence's function):
$$\sum_{n=1}^{\infty} \frac{x^n}{n^2}, \;\; x \in (-\infty, 1]$$
Also it is known that:
$$\sum_{n=1}^{\infty} \frac{\cos n x}{n^2}= ...
- 6,686
8
votes
3
answers
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Prove $\int_{\frac{\pi}{20}}^{\frac{3\pi}{20}} \ln \tan x\,\,dx= - \frac{2G}{5}$
Context:
This question
asks to calculate a definite integral which turns out to be equal to $$\displaystyle 4 \, \text{Ti}_2\left( \tan \frac{3\pi}{20} \right) -
4 \, \text{Ti}_2\left( \tan \frac{\pi}{...
- 6,645
7
votes
3
answers
615
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Closed-forms for the integral $\int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$?
(This is related to this question.)
Define the integral,
$$I_n = \int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$$
with polylogarithm $\operatorname{Li}_n(x)$. Given the Nielsen generalized polylogarithm ...
- 54.9k
6
votes
1
answer
765
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How to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$ using real methods?
How to calculate
$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$$
and
$$\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$$
by means of real methods?
This question was suggested by Cornel the author of the ...
- 25.8k
6
votes
3
answers
691
views
How to find $\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}$ using real analysis and in an elegant way?
I have already evaluated this sum:
\begin{equation*}
\sum_{n=1}^{\infty}\frac{H_nH_{2n}}{n^2}=4\operatorname{Li_4}\left( \frac12\right)+\frac{13}{8}\zeta(4)+\frac72\ln2\zeta(3)-\ln^22\zeta(2)+\frac16\...
- 25.8k
1
vote
1
answer
435
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An integral involving a Gaussian, error functions and the Owen's T function.
This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian .
Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...
- 11.5k
25
votes
4
answers
1k
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Evaluate $\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta $
Evaluate
$$\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\theta $$
Several days ago,I found this interesting integral from a paper about generalized log-sine ...
- 5,281
21
votes
3
answers
2k
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Evaluate $\int_0^1\arcsin^2(\frac{\sqrt{-x}}{2}) (\log^3 x) (\frac{8}{1+x}+\frac{1}{x}) \, dx$
Here is an interesting integral, which is equivalent to the title
$$\tag{1}\int_0^1 \log ^2\left(\sqrt{\frac{x}{4}+1}-\sqrt{\frac{x}{4}}\right) (\log ^3x) \left(\frac{8}{1+x}+\frac{1}{x}\right) \, dx =...
- 19.1k
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
17
votes
2
answers
834
views
Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$
I would like to seek your assistance in computing the sum
$$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$
I am stumped by this sum. I have tried summing the residues of $\displaystyle f(z)=\frac{\pi\...
- 5,604
16
votes
5
answers
1k
views
Double Euler sum $ \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} $
I proved the following result
$$\displaystyle \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} =- \frac{97}{12} \zeta(6)+\frac{7}{4}\zeta(4)\zeta(2) + \frac{5}{2}\zeta(3)^2+\frac{2}{3}\zeta(2)^3$$
After ...
- 14.4k
15
votes
3
answers
956
views
Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$
How to evaluate $$\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx\ ?$$
where $\displaystyle\operatorname{Li}_3(x)=\sum_{n=1}^\infty\frac{x^n}{n^3}$ , $|x|\leq1$
I came across this integral ...
- 25.8k
15
votes
3
answers
2k
views
Simplification of an expression containing $\operatorname{Li}_3(x)$ terms
In my computations I ended up with this result:
$$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102
\...
- 5,342
14
votes
6
answers
3k
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Inverse of the polylogarithm
The polylogarithm can be defined using the power series
$$
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
$$
Contiguous polylogs have the ladder operators
$$
\operatorname{Li}_{s+1}(z) ...
- 1,136
14
votes
5
answers
847
views
Integral $\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$
I would like to know how to evaluate the integral
$$\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$$
I tried expanding the integrand as a series but made little progress as I do not know how to ...
- 5,604
13
votes
2
answers
522
views
On the integral $\int_{0}^{1/2}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz$
This questions is related to my previous one.
I am interested in a explicit evaluation in terms of Euler sums for
$$ \int_{0}^{\pi/4}\text{Li}_3(\cos^2\theta)\,d\theta = \frac{1}{2}\int_{0}^{1/2}\...
- 355k
12
votes
1
answer
611
views
More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$
In this post, the OP asks about the integral,
$$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$
I. User DavidH gave a beautiful (albeit long)...
- 54.9k
11
votes
6
answers
478
views
Prove $\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$
I stumbled upon the interesting definite integral
\begin{equation}
\int\limits_0^1 \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2
\end{equation}
Here is my proof of this result.
Let $u=\sin^{-1}(x)$ ...
- 2,428
11
votes
3
answers
481
views
Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$
While I've studied integrals involving polylogarithm functions I've observed that
$$\int_0^1 \operatorname{Li}_p(x) \, dx \stackrel{?}{=} \sum_{k=2}^p(-1)^{p+k}\zeta(k)+(-1)^{p+1},\tag{1}$$
for any ...
- 12.4k
10
votes
1
answer
991
views
Computing $\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{x}dx$ or $\sum_{n=1}^\infty\frac{H_n^{(4)}}{n2^n}$
Challenging Integral:
\begin{align}
I=\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{x}dx&=6\operatorname{Li}_5\left(\frac12\right)+6\ln2\operatorname{Li}_4\left(\frac12\right)-\frac{81}{16}\zeta(5)-\frac{21}{...
- 25.8k
9
votes
4
answers
596
views
Quadirlogarithm value $\operatorname{Li}_4 \left( \frac{1}{2}\right)$
Is there a known closed form for the following
$$\operatorname{Li}_4 \left( \frac{1}{2}\right)$$
I know that we can derive the closed of $\operatorname{Li}_1 \left( \frac{1}{2}\right),\operatorname{...
- 14.4k
9
votes
2
answers
1k
views
Polylogarithms of negative integer order
The polylogarithms of order $s$ are defined by
$$\mathrm{Li}_s (z) = \sum_{k \geqslant 1} \frac{z^k}{k^s},
\quad |z| < 1.$$
From the above definition, derivatives for the polylogarithms ...
- 11.8k
8
votes
7
answers
907
views
Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$
How would you solve the following
$$\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$$
I might be able to relate the integral to Euler sums .
- 14.4k
8
votes
2
answers
1k
views
Evaluate $\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx$
How to prove $\ \displaystyle\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx=\frac{3\pi^3}{64}+\frac{\pi}{16}\ln^22-\Im\operatorname{Li}_3(1+i)$
Where $\ \displaystyle\operatorname{Li}_3(x)=\sum_{n=1}^\infty\...
- 25.8k
8
votes
2
answers
428
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 ...
- 104k