All Questions
Tagged with polylogarithm logarithms
70
questions
-1
votes
1
answer
199
views
intersection between exponential and polylogarithmic functions
It's possible to solve this equation without using Lambert function or any numerical method, but only with ordinary algebra?
$n^{k}lg_2(n) \le k^n$ with $k,n>0, k \in \mathbb{R}$
For $k=\frac{4}{...
- 23
2
votes
0
answers
74
views
How can I prove that the following function is increasing in $x \in [0,1]$?
How can I prove that the following function is increasing in $x$: $$\sum_{i=1}^{\infty} x (1-x) ^ {i-1} \log \left (1+ \mu (1-x)^{i-1} \right)$$
where $\mu$ is any non-negative number and $x$ is in $[...
- 21
0
votes
2
answers
150
views
Complex logarithms when computing real-valued integral
My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral
\begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx
\end{equation}
...
- 138
8
votes
3
answers
342
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Integral $\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$
It's a follow-up to my previous question.
Can we find an anti-derivative
$$\int\arcsin x\cdot\ln^3x\,dx$$
or, at least, evaluate the definite integral
$$\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$$
in a ...
- 1,928
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:
$$\int\limits_0^{\pi/3}\log(1+\sin x)\log(1-\...
- 2,602
10
votes
2
answers
509
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Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$
I experimentally discovered the following conjectures:
$$\Re\Big[1800\operatorname{Li}_3(i\,\phi)-24\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=100\ln^3\phi-47\,\pi^2\ln\phi-...
- 47.3k
3
votes
2
answers
246
views
closed form iterated logarithms
Is there any way that we could bound the following sum by a closed form expression
$\sum_{i=1}^{\log^* N} \log^{(i)}N$ where $\log^{(i)}$ is the $\log$ function iterated $i$ time?
Thanks
0
votes
2
answers
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Solve for $x,y$: $(3x)^{\log3} = (4y)^{\log4}$ and $4^{\log x} = 3^{\log y}$. [closed]
if $(3x)^{\log3} = (4y)^{\log4}$ and $4^{\log x} = 3^{\log y}$,
then how do I solve for x?
I tried taking log on both sides but after few steps I got stuck
- 41
16
votes
2
answers
888
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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
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
20
votes
1
answer
603
views
A conjectured identity for tetralogarithms $\operatorname{Li}_4$
I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity:
$$720 \,\text{Li}_4\!\left(\tfrac{1}{2}\right)-2160 \,\text{Li}_4\!\left(\tfrac{1}{3}\right)+2160 \,\text{Li}...
- 47.3k
0
votes
2
answers
118
views
Converting an integrand into a polylog?
Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$
I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
- 2,850
1
vote
1
answer
235
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Polylogarithms and the shuffle algebra
$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable
changes of variables.
Attempt: The di-log is defined as $$\...
- 2,850
31
votes
3
answers
2k
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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
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