All Questions
Tagged with polylogarithm logarithms
20
questions with no upvoted or accepted answers
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
$$\int_0^...
- 1,675
6
votes
0
answers
182
views
Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?
This was inspired by Oksana's post. Let, $$a = \ln 2 \quad\quad\\ b = \ln 3\quad\quad\\ c = \ln 5\quad\quad$$
then the following,
\begin{align}
A &= \text{Li}_3\left(\frac12\right)\\
B &= \...
- 54.9k
4
votes
0
answers
341
views
How to evaluate $\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$
I am trying to evaluate
$$\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$$
But I am not sure what to do since integration by parts is not possible here.
I tried using a ...
user809806
4
votes
2
answers
104
views
"Multi-stage logarithm" series expansion (e.g. $a^x+b^x+c^x=d$)
As everyone knows, the solution to $a^x=b$ is $x=\log_a{b}$.
(Edit: Corrected from $x=\log_b{a}$.)
But what about $a^x+b^x=c$?
Let's define a "multilogarithm" function as:
$a_0^x+a_1^x+...+a_n^x=...
user301661
3
votes
0
answers
186
views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \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\...
- 1,675
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
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 424
2
votes
0
answers
74
views
How can I prove that the following function is increasing in $x \in [0,1]$?
How can I prove that the following function is increasing in $x$: $$\sum_{i=1}^{\infty} x (1-x) ^ {i-1} \log \left (1+ \mu (1-x)^{i-1} \right)$$
where $\mu$ is any non-negative number and $x$ is in $[...
- 21
1
vote
0
answers
150
views
Evaluating the indefinite integral $\int x^k \log (1-x) \log (x) \log (x+1) \, dx$
Recently I have calculated the long resisting indefinite integral $\int \frac{1}{x} \log (1-x) \log (x) \log (x+1) \, dx$ (https://math.stackexchange.com/a/3535943/198592).
Similar cases, but for ...
- 12.7k
1
vote
0
answers
74
views
Integrate $\int_{-\infty}^\infty [4(\log r_1 - \log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$
As the title suggests, I am having trouble evaluating the following definite integral:
$$\int_{-\infty}^\infty \left[4\left(\log r_1 - \log r_2\right) - 2\left(\frac{x_1^2}{r_1^2} - \frac{x_2^2}{r_2^...
- 11
1
vote
0
answers
66
views
Multi-logarithm generalisation with multipliers
I previously mentioned
a proposed "multi-stage logarithm" function, and managed to come up with a generalisation of the function.
Originally, the multi-logarithm was defined as:
$a_0^x+a_1^x+...+...
user301661
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
0
votes
1
answer
436
views
What's a better time complexity, $O(\log^2(n))$, or $O(n)$?
(By $O(\log^2(n))$, I mean $O((\log n)^2)$ rather than $O(\log(\log(n)) )$
I know $O(\log(n))$ is better than $O(\log^2(n))$ which itself is better than $O(\log^3(n))$ etc. But how do these compare to ...
0
votes
0
answers
59
views
Stirling number series resummation
\begin{equation}\sum_{m=1}^{\infty}\frac{a_1^3 S_m^{(3)} (u-1)^{m-1}
\left(\frac{x}{x-1}\right)^m}{m!}\end{equation}
Does somebody know the result of this resummation?
Note:
$S_m^{(3)} $ belongs to ...
- 99
0
votes
0
answers
82
views
General expression of a triangle sequence
\begin{gather*}
\frac{1}{4} \\
\frac{1}{4} \quad \frac{1}{4} \\
\frac{11}{48} \quad \frac{1}{4} \quad \frac{11}{48} \\
\frac{5}{24} \quad \frac{11}{48} \quad \frac{11}{48} \quad \frac{5}{24} \\
\frac{...
- 99
0
votes
0
answers
50
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Further question on Logarithm product integral
How to perform $\int_0^1 \frac{\left(a_0\log(u)+a_1\log(1-u)+a_{2}\log(1-xu)\right)^9}{u-1} du $?
Method tried:
Intgration-by-parts
Series expansion
change of variable $\log(u)=x$
But I still can't ...
- 99