Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
546
questions
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$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
0
votes
0
answers
88
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$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 \operatorname{Li}_{2}\left(\frac12 \...
4
votes
0
answers
123
views
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 ...
0
votes
0
answers
34
views
Asymptotic expansions of sums via Cauchy's integral formula and Watson's lemma
I am looking for references that deal with the asymptotic expansions of sums of the form
$$s(n)=\sum_{k=0}^n g(n,k)$$
using the (or similar to) following method.
We have the generating function
$$f(z)=...
3
votes
1
answer
56
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
2
votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
4
votes
1
answer
224
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Closed form for $\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(n+1)^2 \Gamma(M-n)\Gamma(n+1)}$
I encountered this expression generated by mathematica as a sub-step in a problem I am solving.
$$\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(...
0
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1
answer
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$Im[\log(\frac{3}{2})Li_{2}(\frac{3}{2})-Li_{3}(\frac{3}{2})+2Li_{3}(3)]=-\frac{\pi}{2}\log^{2}(2)-\frac{3\pi}{2}\log^{2}(3)+\pi\log(2)\log(3)$
When working on another problem, I got the following expression
\begin{align}
& \Im\left[\log\left(\frac32\right) \operatorname{Li}
_{2} \left(\frac32\right) - \operatorname{Li}
_{3}\left(\frac32\...
3
votes
1
answer
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Is there a closed form solution for the sum $\sum\limits_{M=2}^{\infty} \sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}}{(M-n-1)(n+1)^{2}}$?
While working on another problem, this came up as a sub-step:
$$
\sum_{M\ =\ 2}^{\infty}\,\,\,\sum_{n\ =\ M}^{\infty}\
{2^{-M}\ 3^{M - n - 1} \over \left(M - n - 1\right)\left(n + 1\right)^{\,2}}
$$
...
1
vote
0
answers
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How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
9
votes
0
answers
254
views
Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
1
vote
0
answers
80
views
Simplify the Laplace Transform for $E_{i}(-y)^{2}$
I want to simplify the Laplace transform expression of $E_{i}(-y)^{2}$, where $E_{i}(y)$ is the exponential integral defined by $E_{i}(y) = -\int\limits_{-y}^{\infty} \frac{e^{-t}}{t} dt$.
Question: ...
1
vote
0
answers
50
views
Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$
I am now trying a direct approach to solving my question about
$$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$
where the $a_i$ are all positive. Note that the $\arctan$s ...
8
votes
1
answer
285
views
Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$
with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$
$$I(a,b)=
\frac\pi4\left(\frac{\pi^2}6
-\Li\...
12
votes
2
answers
499
views
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
3
votes
1
answer
69
views
Behaviour of polylogarithm at $|z|=1$
I have the sum
$$
\sum_{n=1}^\infty \dfrac{\cos (n \theta)}{n^5} = \dfrac{\text{Li}_5 (e^{i\theta}) + \text{Li}_5 (e^{-i\theta})}{2},
$$
where $0\leq\theta < 2 \pi$ is an angle and $\text{Li}_5(z)$ ...
0
votes
0
answers
50
views
How to integrate $\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}$?
I am trying to compute the integral
$$\int_{x_0}^{1}\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}\text{d}x$$
where $x_0, x_1$ and $x_2$ are related to some parameters $\kappa_\pm$ by
$$x_0=\...
4
votes
0
answers
124
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Is it possible to evaluate this integral? If not, is it possible to determine whether the result is an elliptic function or not?
I am trying to evaluate the integral
$$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-...
0
votes
0
answers
39
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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
57
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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 ...
3
votes
0
answers
186
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
$$ B_n=\sum_{k=0}^{\lfloor\...
2
votes
0
answers
68
views
Help verifying expression involving dilogarithms.
I need help verifying that the following equality holds:
$$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)$$
$$=$$
$$\...
4
votes
3
answers
136
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I need help evaluating the integral $\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$
I was playing around with the integral: $$\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$$
I couldn't find a way of solving it, but I used WolframAlpha to find that the integral evaluated ...
0
votes
1
answer
38
views
Upper bound of a polylogarithm type series
The series is
$\sum_{n=0}^{\infty}\left(1+\frac{n}{a}\right)^bx^n,$
where $a$ and $b$ are positive real numbers, $x\in[0,1]$.
The sum of the series diverges when $x\to1$. I want to get an upper bound ...
21
votes
1
answer
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Solution of a meme integral: $\int \frac{x \sin(x)}{1+\cos(x)^2}\mathrm{d}x$
Context
A few days ago I saw a meme published on a mathematics page in which they joked about the fact that $$\int\frac{x\sin(x)}{1+\cos(x)^2}\mathrm{d}x$$ was very long (and they put a screen shot of ...
2
votes
2
answers
154
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$\displaystyle\int_{0}^{\frac{\pi}{2}}\ln(1+\alpha^N\tan(x)^N)\mathrm{d}x\quad$ where $N\in\mathbb{N}$
$\color{red}{\textrm{Context}}$
I wanted to calculate the following integrals
$$\displaystyle\int_{0}^{\frac{\pi}{2}}\ln(1+\tan(x)^N)\mathrm{d}x\qquad\text{for }N\in\mathbb{N}$$
and I used the Feymann ...
1
vote
0
answers
59
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Contour integration with polylogarithm
Starting from Bose-Einstein integral representation of the polylogarithm
$$Li_{s}(z) = \frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t/z - 1}dt \quad \quad(1)$$
it's not too hard to obtain the ...
3
votes
0
answers
40
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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(...
2
votes
1
answer
181
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Approximation for $\sum_{i=1}^\infty \frac{\sqrt i}{\sqrt{2\pi \epsilon}} e^{\frac{-i}{2\epsilon}}$?
I am trying to get some nice number (not the polylogarithmic) with the series on the title. I know the exact result is $\dfrac{1}{\sqrt{2\pi \epsilon}}Li_{-\frac{1}{2}}(\dfrac{1}{^{\epsilon}\sqrt{e}})$...
2
votes
0
answers
129
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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{\...