All Questions
Tagged with polylogarithm logarithms
70
questions
41
votes
2
answers
2k
views
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:
$$I\approx-0....
- 47.3k
37
votes
2
answers
4k
views
A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$
A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
- 13.2k
33
votes
4
answers
3k
views
Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$
This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here:
Prove:
$$\int_0^1\ln(1-x)\ln(1+x)\ln^...
- 5,342
31
votes
3
answers
2k
views
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
29
votes
4
answers
10k
views
Yet another log-sin integral $\int\limits_0^{\pi/3}\log(1+\sin x)\log(1-\sin x)\,dx$
There has been much interest to various log-trig integrals on this site (e.g. see [1][2][3][4][5][6][7][8][9]).
Here is another one I'm trying to solve:
$$\int\limits_0^{\pi/3}\log(1+\sin x)\log(1-\...
- 2,602
21
votes
1
answer
921
views
Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$
This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$.
Is there a closed form for this integral?
$$I=\int_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx\...
- 47.3k
20
votes
1
answer
603
views
A conjectured identity for tetralogarithms $\operatorname{Li}_4$
I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity:
$$720 \,\text{Li}_4\!\left(\tfrac{1}{2}\right)-2160 \,\text{Li}_4\!\left(\tfrac{1}{3}\right)+2160 \,\text{Li}...
- 47.3k
16
votes
2
answers
888
views
Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$
I'm interested in the following definite integral:
$$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$
The corresponding antiderivative can be evaluated with Mathematica, but even after ...
- 47.3k
16
votes
1
answer
551
views
Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points
The polylogarithm is defined by the series
$$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$
There are relations connecting values of the polylogarithm at certain rational points in the ...
- 47.3k
15
votes
3
answers
2k
views
Simplification of an expression containing $\operatorname{Li}_3(x)$ terms
In my computations I ended up with this result:
$$\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
views
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
$$
\operatorname{Li}_{s+1}(z) ...
- 1,136
13
votes
2
answers
287
views
Log integrals I
In this example the value of the integral
\begin{align}
I_{3} = \int_{0}^{1} \frac{\ln^{3}(1+x)}{x} \, dx
\end{align}
was derived. The purpose of this question is to determine the value of the more ...
- 26.6k
12
votes
3
answers
460
views
How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$
I am trying evaluating this
$$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\ \text{d}x.$$
For $k=1$, there has
$$J(1)=\frac{\pi^4}{96}.$$
Maybe $J(k)$ ...
- 5,370
11
votes
1
answer
255
views
A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$
I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
- 1,724
11
votes
0
answers
255
views
Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$
I tried to solve this integral and got it, I showed firstly
$$\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^...
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