Skip to main content

Questions tagged [polylogarithm]

For questions about or related to polylogarithm functions.

Filter by
Sorted by
Tagged with
3 votes
1 answer
93 views

Integrate $\frac1{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$ over $[-1,1]^2$?

Problem $$ \text{Evaluate} \quad I = \iint\limits_{[-1,1]^2} \frac{du\,dv}{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$$ The integral is a special case of one that appeared in this recent post, in which ...
user170231's user avatar
  • 20.6k
3 votes
3 answers
386 views

$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ 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^{\...
Srini's user avatar
  • 862
0 votes
0 answers
96 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 \...
Srini's user avatar
  • 862
4 votes
0 answers
125 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 ...
popi's user avatar
  • 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)=...
bob's user avatar
  • 2,249
3 votes
1 answer
62 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 ...
Jessie Christian's user avatar
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 ...
Jack's user avatar
  • 424
4 votes
1 answer
227 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)}{(...
Srini's user avatar
  • 862
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\...
Srini's user avatar
  • 862
3 votes
1 answer
117 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: $$ \sum_{M\ =\ 2}^{\infty}\,\,\,\sum_{n\ =\ M}^{\infty}\ {2^{-M}\ 3^{M - n - 1} \over \left(M - n - 1\right)\left(n + 1\right)^{\,2}} $$ ...
Srini's user avatar
  • 862
1 vote
0 answers
67 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+...
Mods And Staff Are Not Fair's user avatar
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, ...
Setness Ramesory's user avatar
1 vote
0 answers
81 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: ...
Srini's user avatar
  • 862
1 vote
0 answers
51 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 ...
Parcly Taxel's user avatar
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\...
Parcly Taxel's user avatar
12 votes
2 answers
504 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 ...
Alma Arjuna's user avatar
  • 3,881
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 $\text{Li}_5(z)$ ...
Gabriel Ybarra Marcaida's user avatar
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=\...
Anders W's user avatar
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-...
Pxx's user avatar
  • 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 ...
MultipleSearchingUnity's user avatar
1 vote
1 answer
59 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 ...
Dr Potato's user avatar
  • 812
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\...
Faoler's user avatar
  • 1,647
2 votes
0 answers
68 views

Help verifying expression involving dilogarithms.

I need help verifying that the following equality holds: $$Li_2(-2-2\sqrt2)+Li_2(3-2\sqrt2)+Li_2(\frac{1}{\sqrt2})-Li_2(-\frac{1}{\sqrt2})-Li_2(2-\sqrt2)-Li_2(-1-\sqrt2)-2Li_2(-3+2\sqrt2)$$ $$=$$ $$\...
Noa Arvidsson's user avatar
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 ...
Abdullah's user avatar
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 ...
Jobs Adam's user avatar
  • 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 ...
Math Attack's user avatar
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 ...
Math Attack's user avatar
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(...
Math Attack's user avatar
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 $\dfrac{1}{\sqrt{2\pi \epsilon}}Li_{-\frac{1}{2}}(\dfrac{1}{^{\epsilon}\sqrt{e}})$...
Federico's user avatar
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{\...
Math Attack's user avatar
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, ...
James Arathoon's user avatar
1 vote
1 answer
112 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 ...
QuantumLogarithm's user avatar
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 ...
Artur Wiadrowski's user avatar
0 votes
3 answers
82 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 ...
Kraken's user avatar
  • 27
11 votes
1 answer
254 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 ...
Dylan Levine's user avatar
  • 1,688
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}-\...
garondal's user avatar
  • 889
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^...
Faoler's user avatar
  • 1,647
4 votes
0 answers
112 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-...
Dr. Wolfgang Hintze's user avatar
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 ...
Setness Ramesory's user avatar
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) - {\...
OnTheWay's user avatar
  • 2,702
1 vote
0 answers
69 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 ...
Artur Wiadrowski's user avatar
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)}\...
David H's user avatar
  • 30.7k
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}...
Math Attack's user avatar
2 votes
1 answer
75 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 ...
Po1ynomial's user avatar
  • 1,695
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 ...
Rudyard's user avatar
  • 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}\,...
Person's user avatar
  • 1,123
1 vote
1 answer
202 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 ...
Artur Wiadrowski's user avatar
6 votes
2 answers
326 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 ...
Ali Shadhar's user avatar
  • 25.8k
1 vote
0 answers
86 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: $$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
user3108815's user avatar
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}...
Max's user avatar
  • 910

1
2 3 4 5
11