Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
113
questions with no upvoted or accepted answers
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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
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
- 25.8k
11
votes
0
answers
255
views
Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$
I tried to solve this integral and got it, I showed firstly
$$\int_0^1 \frac{\ln^2(x+1)+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx=2\Im\left[\text{Li}_3(1+i) \right] $$
and for other integral
$$\int_0^...
- 1,637
11
votes
0
answers
436
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
$$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
- 25.8k
10
votes
0
answers
259
views
Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
- 5,370
8
votes
0
answers
404
views
Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$
This integral can be found in Cornel's book, (Almost) Impossible Integral, Sums and Series page $97$ where he showed that
$$\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt=\frac14\left(\frac12\ln^2(1+x^2)-2\...
- 25.8k
8
votes
0
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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,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
8
votes
0
answers
295
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Why does the tribonacci constant have a trilogarithm ladder?
When I came across the dilogarithm ladders of Coxeter and Landen, namely,
$$\text{Li}_2\Big(\frac1{\phi^6}\Big)-4\text{Li}_2\Big(\frac1{\phi^3}\Big)-3\text{Li}_2\Big(\frac1{\phi^2}\Big)+6\text{Li}_2\...
- 54.9k
6
votes
1
answer
285
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Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$
this integral got posted on a mathematics group by a friend
$$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$
I tried seeing what I'd get from ...
- 231
6
votes
0
answers
362
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Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 5,370
6
votes
0
answers
306
views
Does there exist a closed form for $\int_0^{\pi/2}\frac{x^2\ \text{Li}_2(\sin^2x)}{\sin x}dx$?
I am not sure if there exists a closed form for
$$I=\int_0^{\pi/2}\frac{x^2\ \text{Li}_2(\sin^2x)}{\sin x}dx$$
which seems non-trivial.
I used the reflection and landen's identity, didn't help much.
...
- 25.8k
6
votes
0
answers
182
views
Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?
This was inspired by Oksana's post. Let, $$a = \ln 2 \quad\quad\\ b = \ln 3\quad\quad\\ c = \ln 5\quad\quad$$
then the following,
\begin{align}
A &= \text{Li}_3\left(\frac12\right)\\
B &= \...
- 54.9k
6
votes
0
answers
197
views
Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$
I am trying to evaluate the following integral
$$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0<...
- 93
5
votes
0
answers
281
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Is there a closed form without MZV for $ \sum _{k=1}^{\infty }\frac{H_k}{k^6\:2^k}$?
While evaluating the weight $7$ integral $\displaystyle \int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\right)}{1+x}\:dx$
I managed to prove that
$$\int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\...
- 2,646
5
votes
0
answers
161
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Verifying closed form evaluation of an Ising-class multiple integral
For $n\in\mathbb{N}\land n\ge2$, define the so-called Ising-class integral of the third kind, $E_{n}$, via the sequence of $\left(n-1\right)$-dimensional integrals
$$E_{n}:=2\int_{\left[0,1\right]^{...
- 30.7k
4
votes
0
answers
125
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Definite integral involving exponential and logarith function
Working with Dilogarimth function, we get the following definite integral
$$\int_0^{\infty}\frac{t^2\,\ln^{n}(t)}{(1-e^{x\,t})(1-e^{y\,t})}\,dt$$
with $n=1,2,3,...$ and $x,y>0$.
I wonder if is ...
- 1,774
4
votes
0
answers
124
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Is it possible to evaluate this integral? If not, is it possible to determine whether the result is an elliptic function or not?
I am trying to evaluate the integral
$$F(x,y) = \int_0^1 du_1\, \int_0^{1-u_1} du_2\, \frac{\log f(x,y|u_1,u_2)}{f(x,y|u_1,u_2)}\,, \tag{1}$$
with
$$f(x,y|u_1,u_2) := u_1(1-u_1)+y\, u_2(1-u_2) + (x-y-...
- 697
4
votes
0
answers
112
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Calculate an integral involving polylog functions
Im my recent answer https://math.stackexchange.com/a/4777055/198592 I found numerically that the following integral has a very simple result
$$i = \int_0^1 \frac{\text{Li}_2\left(\frac{i\; t}{\sqrt{1-...
- 12.7k
4
votes
0
answers
83
views
Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,702
4
votes
0
answers
81
views
How to derive this polylogarithm identity (involving Bernoulli polynomials)?
How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
- 6,692
4
votes
0
answers
220
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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
$$ \frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right) + \...
- 2,017
4
votes
0
answers
341
views
How to evaluate $\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$
I am trying to evaluate
$$\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$$
But I am not sure what to do since integration by parts is not possible here.
I tried using a ...
user809806
4
votes
0
answers
74
views
Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?
Let us define a following function:
\begin{eqnarray}
{\mathcal J}^{(d)}(\vec{A}) := \int\limits_0^\infty e^{-u^2} \prod\limits_{\xi=1}^d erf(A_\xi u) du
\end{eqnarray}
for $\vec{A}:=\left(A_\xi\right)...
- 11.5k
4
votes
2
answers
104
views
"Multi-stage logarithm" series expansion (e.g. $a^x+b^x+c^x=d$)
As everyone knows, the solution to $a^x=b$ is $x=\log_a{b}$.
(Edit: Corrected from $x=\log_b{a}$.)
But what about $a^x+b^x=c$?
Let's define a "multilogarithm" function as:
$a_0^x+a_1^x+...+a_n^x=...
user301661
4
votes
0
answers
78
views
An anti-derivative of a product of three poly-logarithms and a simple fraction.
It is fairly well known that integrating a function that involves products of polylogarithms is not always possible. The motivation for this question is to reduce a large class of such integrals to ...
- 11.5k
4
votes
0
answers
301
views
Simplification of an expression involving the dilogarithm with complex argument
Do you think there is a way to get a nice form of the expression below
$$\Im{\left( \text{Li}_2\left(\frac{3}{5}+\frac{4 i}{5}\right)- \text{Li}_2\left(-\frac{3}{5}+\frac{4 i}{5}\right)+ \text{Li}_2\...
- 44.4k
3
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0
answers
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views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\...
- 1,637
3
votes
0
answers
40
views
Regularization involving Stieltjes constants: $\displaystyle\sum_{k=1}^{\infty}\frac{\ln(k)^n}{k}\overset{\mathcal{R}}{=}\gamma_n$
Notation
$\zeta(z)$ is the Riemann zeta function
$\operatorname{Li}_{\nu}(z)$ is the polylogarithm function
$\operatorname{Li}^{(n,0)}_{\nu}(z):=\frac{\partial^n}{\partial\nu^n}\operatorname{Li}_\nu(...
3
votes
0
answers
121
views
Show that $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$ (polylogarithm)
I am working with the polylogarithm function and want to find closed expressions for $\textrm{Li}_2(e^{ix})$.
If I plot the function $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))$ I get $y=\dfrac{x^2}{4}-\...
- 889
3
votes
0
answers
142
views
Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$
Prove that $$-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$$
where ${\rm Li}_2(x)$ is the Poly Logarithm function and $\zeta(s)$ is the Riemann zeta function
Let $$I=-\int_{0}^{1}...
- 910
3
votes
0
answers
316
views
Two tough integrals with logarithms and polylogarithms
The following two integrals are given in (Almost) Impossible Integrals, Sums, and Series (see Sect. $\textbf{1.55}$, page $35$),
$$i) \int_0^{\pi/2} \cot (x) \log (\cos (x)) \log ^2(\sin (x)) \...
- 5,495
3
votes
0
answers
77
views
Asymptotics of Geometric Distribution Moments
It is known (e.g., see Wikipedia) that the $k$th moment of a Geometric distribution with success probability $p$ is
$$\mathbb{E}\left[X^k\right] = \sum_{j = 0}^\infty j^k \cdot p(1-p)^j = p \cdot\text{...
- 6,603
3
votes
0
answers
291
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
$$
-\frac{1}{2}\int_{0}^{1}\ln^2t
\int_{1}^{\infty}\frac{x\ln(x-1)}{(1+tx)(1+x^2)}\text{d}x\text{...
- 5,370
3
votes
0
answers
184
views
How to simplify this polylog expression $\operatorname{Li}_4\left(\frac{z-1}{z}\right)$?
Evaluating this integral in Mathematica
$$i_2(z)=\int_0^z \frac{\log ^2(x) \log (1-x)}{1-x} \, dx\tag{1}$$
returns a mixture of polylogs up to order 4 and several log-terms.
In the region of ...
- 12.7k
3
votes
0
answers
75
views
Approaching a branch point along different paths
There's a very nice characterization of the three main types of isolated singularities of an analytic function $f(z)$ that takes oriented curves $\gamma$ that terminate at the singularity and ...
- 6,280
3
votes
0
answers
106
views
Proving that swapping the order of this summation is justified
I'm unsure if this has been discovered already, but it's heavily related to my current research, particularly to this question of mine (this conjecture was also originally posted at the beginning of ...
- 856
3
votes
0
answers
111
views
What is the integral of the square of a general polylogarithm with gaussian argument?
I'm attempting an integration of such a function. Mathematica won't help me and I've tried several series decompositions which haven't yielded much yet.
So I'm after: $\int_{-\infty}^{\infty}\mathrm{...
- 31
2
votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 424
2
votes
0
answers
68
views
Help verifying expression involving dilogarithms.
I need help verifying that the following equality holds:
$$Li_2(-2-2\sqrt2)+Li_2(3-2\sqrt2)+Li_2(\frac{1}{\sqrt2})-Li_2(-\frac{1}{\sqrt2})-Li_2(2-\sqrt2)-Li_2(-1-\sqrt2)-2Li_2(-3+2\sqrt2)$$
$$=$$
$$\...
- 359
2
votes
0
answers
131
views
Generating function of Clausen functions: $\displaystyle\sum_{n=1}^\infty \text{Cl}_{2n}(x)\frac{t^{2n}}{(2n)!}$
Context
I was trying to solve a series:
$$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$
Using the Fourier series
\begin{equation}
\begin{split}
\log\Gamma(x)&=\frac{\ln(2\pi)}{2}+\sum_{k=1}^\infty \frac{\...
2
votes
0
answers
140
views
The ultimate polylogarithm ladder
As you can see, here I performed a derivation of a quite simple formula, not much differing from the standard integral representation of the Polylogarithm. Seeking to make it fancier, I arrived at ...
- 1,037
2
votes
0
answers
85
views
Complex polylogarithm/Clausen function/Fourier series
Sorry for the confusing title but I'm having a problem and I can phrase the question in multiple different ways.
I was calculating with WolframAlpha
$$\int \text{atanh}(\cos(x))\mathrm{d}x= i \text{Li}...
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.
$$\int\frac{\log(x+a)}{x}\,...
- 1,123
2
votes
0
answers
36
views
Canonical reference for algebraic theory of polylogs?
I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to ...
2
votes
1
answer
138
views
Powers of polylogarithms
I would like to take powers of arbitrary order to polylogarithm functions. For instance, given
$$
\text{Li}_\alpha(z) = \sum_{k=1}^\infty \frac{z^k}{k^\alpha}
$$
I am interested in
$$
[\text{Li}_\...
- 551
2
votes
1
answer
326
views
Evaluate $\Im(\operatorname{Li}_3(2i) + \operatorname{Li}_3(\frac i2))$
Applying the trilogarithm identity
$$ \operatorname{Li}_{3}\left(z\right) - \operatorname{Li}_{3}\left(1 \over z\right) =
-{1 \over 6}\ln^{3}\left(-z\right) -
{\pi^{2} \over 6}\ln\left(-z\right)\tag{1}...
- 99.7k
2
votes
0
answers
92
views
An integral involving a Gaussian and a power of a normal cumulative distribution function
Being inspired by How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$? we formulated the question below.
Let $c \in (0,1/\sqrt{2})$ and let $n \in \Bbb{N}$. Then let $\phi(x) : =\frac{\...
- 11.5k
2
votes
0
answers
142
views
Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$
I am trying to evaluate
$$I=\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$$
Integration by parts yields
$$I=\frac58\zeta(4)-\frac12\int_0^1\frac{\ln(1-x)\text{Li}_2(-x^2)}{x}dx$$
Another related ...
- 25.8k
2
votes
1
answer
86
views
Is there a nice way to represent $\sum_{n=1}^\infty \frac{(-1)^{n+1}H_n}{n+m+1}$?
Here, $H_n$ denotes the harmonic number. More colloquially, is there any way to represent $$\int_0^1 x^{n-1}\log^2\left(1+x\right)\ \mathrm{d}x$$ in a nice way? The latter is corollary to the original ...
- 125
2
votes
0
answers
49
views
Where am I wrong in my calculations involving the Polylogarithm?
I was messing around with some formulas for the Polylogarithm $\operatorname{Li}_s(z)$ when I got the result that $0 = \frac{\sqrt{\pi}}{2}$ which is clearly absurd. Here are my calculations:
Under ...
- 1,054
2
votes
0
answers
154
views
Evaluating a variant of the polylogarithm
Consider the infinite sum :
$$\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 ...
