Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
546
questions
2
votes
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+100
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 814
0
votes
0
answers
88
views
$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 \operatorname{Li}_{2}\left(\frac12 \...
- 814
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 ...
- 1,774
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
$$f(z)=...
- 2,217
3
votes
1
answer
56
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
- 203
2
votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 376
4
votes
1
answer
224
views
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.
$$\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(...
- 814
0
votes
1
answer
53
views
$Im[\log(\frac{3}{2})Li_{2}(\frac{3}{2})-Li_{3}(\frac{3}{2})+2Li_{3}(3)]=-\frac{\pi}{2}\log^{2}(2)-\frac{3\pi}{2}\log^{2}(3)+\pi\log(2)\log(3)$
When working on another problem, I got the following expression
\begin{align}
& \Im\left[\log\left(\frac32\right) \operatorname{Li}
_{2} \left(\frac32\right) - \operatorname{Li}
_{3}\left(\frac32\...
- 814
3
votes
1
answer
116
views
Is there a closed form solution for the sum $\sum\limits_{M=2}^{\infty} \sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}}{(M-n-1)(n+1)^{2}}$?
While working on another problem, this came up as a sub-step:
$$
\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}}
$$
...
- 814
1
vote
0
answers
64
views
How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\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.
$$
I came across this when playing with Legendre polynomials, ...
- 5,362
1
vote
0
answers
80
views
Simplify the Laplace Transform for $E_{i}(-y)^{2}$
I want to simplify the Laplace transform expression of $E_{i}(-y)^{2}$, where $E_{i}(y)$ is the exponential integral defined by $E_{i}(y) = -\int\limits_{-y}^{\infty} \frac{e^{-t}}{t} dt$.
Question: ...
- 814
1
vote
0
answers
50
views
Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$
I am now trying a direct approach to solving my question about
$$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$
where the $a_i$ are all positive. Note that the $\arctan$s ...
- 104k
8
votes
1
answer
285
views
Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$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\...
- 104k
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 ...
- 3,871
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
$$x_0=\...
- 1
4
votes
0
answers
124
views
Is it possible to evaluate this integral? If not, is it possible to determine whether the result is an elliptic function or not?
I am trying to evaluate the integral
$$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
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
57
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 ...
- 800
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
$$ B_n=\sum_{k=0}^{\lfloor\...
- 1,577
2
votes
0
answers
68
views
Help verifying expression involving dilogarithms.
I need help verifying that the following equality holds:
$$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
4
votes
3
answers
136
views
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 ...
- 71
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 ...
- 243
21
votes
1
answer
1k
views
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
views
$\displaystyle\int_{0}^{\frac{\pi}{2}}\ln(1+\alpha^N\tan(x)^N)\mathrm{d}x\quad$ 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
views
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 ...
- 342
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(...
2
votes
1
answer
181
views
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 $\dfrac{1}{\sqrt{2\pi \epsilon}}Li_{-\frac{1}{2}}(\dfrac{1}{^{\epsilon}\sqrt{e}})$...
- 59
2
votes
0
answers
129
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{\...
0
votes
0
answers
37
views
Generalised Polylogarithm Polynomials and Related Integer Sequences
Consider the generalised infinite summation
$$S_{n,m}=m^{n+1} \sum_{k=1}^\infty \frac{k^n}{(m+1)^k}=m^{n+1}\,\mathrm{Li}_{(-n)} \left(\frac{1}{m+1}\right)$$
where $m$ and $n$ are positive integers, ...
- 3,667
1
vote
1
answer
111
views
asymptotic behaviour of polylogarithmic function
I would like to understand the asymptotic behaviour as $a \rightarrow 0$ of the function
$$
f(a) := \sum\limits_{k=2}^{\infty} e^{ - a^2 k}{k^{-3/2}}
$$
More precisely, I would like to obtain an ...
- 200
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 ...
- 1,039
0
votes
3
answers
80
views
Evaluating an integral from 0 to 1 with a parameter, (and a dilogarithm)
So I need to evaluate the following integral (in terms of a):
$$\int_{0}^{1} \frac{\ln{|1-\frac{y}{a}|}}{y} dy$$
Till now I have tried u-sub ($u = \ln{|1-\frac{y}{a}|}$, $u=\frac{y}{a}$) and ...
- 27
11
votes
1
answer
252
views
A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$
I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
- 1,686
3
votes
0
answers
119
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}-\...
- 857
11
votes
0
answers
252
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
$$\int_0^...
- 1,577
4
votes
0
answers
111
views
Calculate an integral involving polylog functions
Im my recent answer https://math.stackexchange.com/a/4777055/198592 I found numerically that the following integral has a very simple result
$$i = \int_0^1 \frac{\text{Li}_2\left(\frac{i\; t}{\sqrt{1-...
- 12.7k
8
votes
3
answers
1k
views
Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$
Is it possible to show
$$
\int_{0}^{1}\frac{K(k)\ln\left[\tfrac{\left ( 1+k \right)^3}{1-k} \right] }{k}
\text{d}k=\frac{\pi^3}{4}\;\;?
$$
where $K(k)$ is the complete elliptic integral of the first ...
- 5,362
4
votes
0
answers
83
views
Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,702
1
vote
0
answers
68
views
Polylogarithm further generalized
Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
- 1,039
5
votes
1
answer
193
views
Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
2
votes
0
answers
84
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
$$\int \text{atanh}(\cos(x))\mathrm{d}x= i \text{Li}...
2
votes
1
answer
71
views
Converting polylogarithms to Dirichlet L functions
When trying to simplify polylogarithms evaluated at some root of unity, namely $\text{Li}_s(\omega)$ for $\omega=e^{2\pi i ~r/n}$, it is reasonable to convert it to Hurwitz zeta functions or Dirichlet ...
- 1,686
1
vote
1
answer
60
views
Imaginary part of the dilogarithm of an imaginary number
I am wondering if I can simplify
$${\rm Im} \left[ {\rm Li}_2(i x)\right] \ , $$
in terms of more elementary functions, when $x$ is real (in particular, I am interested in $0<x<1$). I checked ...
- 305
2
votes
0
answers
68
views
Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\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
1
vote
1
answer
201
views
Verification of the generalized polylogarithm formula
Here I posted a generalized formula for the polylogarithm I had discovered. However, for $x=\frac{1}{2}$, $z=\frac{1}{2}$, $p=1$ wolfram alpha yields a result different than what the double integral ...
- 1,039
6
votes
2
answers
324
views
How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$
I am trying to prove that
$$\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G,$$
where $G$ is the Catalan constant and $\operatorname{Li}_2(x)$ is the dilogarithm ...
- 25.8k
1
vote
0
answers
81
views
Efficient calculation for Lerch Transcendent Expression
I've encountered:
$$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$
When trying to compute:
$$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
- 301
3
votes
0
answers
141
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}...
- 862