All Questions
Tagged with polylogarithm logarithms
70
questions
3
votes
3
answers
388
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 $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 862
3
votes
1
answer
63
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\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 ...
- 424
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
$$x_0=\...
- 1
3
votes
0
answers
186
views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \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\...
- 1,685
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
$$\int_0^...
- 1,685
0
votes
1
answer
436
views
What's a better time complexity, $O(\log^2(n))$, or $O(n)$?
(By $O(\log^2(n))$, I mean $O((\log n)^2)$ rather than $O(\log(\log(n)) )$
I know $O(\log(n))$ is better than $O(\log^2(n))$ which itself is better than $O(\log^3(n))$ etc. But how do these compare to ...
0
votes
1
answer
34
views
Proving $2^{\log^{1-\varepsilon}n}\in \omega(\operatorname{polylog}(n))$
Let $\varepsilon \in (0,1)$. I wish to show that $2^{\log^{1-\varepsilon}(n)}\in \omega(\operatorname{polylog}(n))$. I attempted to turn this into a function and use L'Hospital's rule but that got me ...
- 2,885
3
votes
1
answer
148
views
Explicit value of $\operatorname{Li}_2(1/2-{\rm i}/2)$
When you ask Wolfram Alpha about the value of $\operatorname{Li}_{2}\left(1/2-{\rm i}/2\right)$ it gives you $$
\frac{5\pi^{2}}{96} -
\frac{\ln^{2}\left(2\right)}{8} +
{\rm i}\left[\frac{\pi\ln\left(2\...
4
votes
1
answer
257
views
Find closed-form of: $\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$
I found this integral: $$\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$$
And it seems look like this problem but i don't know how to process with this one.
First, i tried to use series of $\frac{x}{x^...
- 2,732
0
votes
1
answer
51
views
How is $(2^a)^{\lg n} = n^a$? [closed]
I was learning from Introduction to Algorithms (Chapter 3 under the topic “Logarithms”) and came across this expression.
$$
\lim_{ n \to 0 }{\frac{\lg^b n}{(2^a)^{\lg n}}}
=
\lim_{n \to 0} \frac{...
- 325
7
votes
2
answers
132
views
Series expansion of $\text{Li}_3(1-x)$ at $x \sim 0$
My question is simple, but maybe hard to answer. I would like to have a series expansion for $\text{Li}_3 (1-x)$ at $x \sim 0$ in the following form:
$$\text{Li}_3 (1-x) = \sum_{n=0} c_n x^n + \log x \...
- 697
2
votes
1
answer
217
views
Can this formula for $\zeta(3)$ be proven or simplified further?
This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
- 7,631
0
votes
1
answer
78
views
How to solve a Non-algebraic equation?
While working with exponential growth and decay, I encountered a problem where I need to solve an equation involving logarithm. I could not separate or could not make it explicit.
$y=\frac{\ln((1+r)(1-...
- 180
12
votes
3
answers
460
views
How to evaluate$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
4
votes
3
answers
211
views
Does $\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\right)\,\ln\left(1-\frac{\cos^2\phi}{C^2}\right)$ have a closed form?
I am wondering if anyone has a nice way of approaching the following definite integral $\newcommand{\dilog}{\operatorname{Li}_2}$
$$\int_0^{2\pi}\frac{d\phi}{2\pi} \,\ln\left(\frac{\cos^2\phi}{C^2}\...
- 179
5
votes
2
answers
368
views
Closed form of the sum $s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}$
I am interested to know if the following sum has a closed form
$$s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}\tag{1}$$
I stumbled on this question while studying a very useful book about ...
- 12.7k
4
votes
1
answer
82
views
Evaluate the following series sum.
Problem
I’m trying to evaluate the following series sum
\begin{equation}
S_{j}(z) = \sum_{k=1}^{\infty} \frac{2 H_{k} z^{k+2}}{(k+1)(k+2)^{j}}
\end{equation}
Where
\begin{equation}
H_{k} = \sum_{n=1}^{...
- 87
1
vote
1
answer
83
views
Integral of a modified softplus function
In a manuscript I am currently reading, the authors propose a modified softplus function
$$g(a)=\frac{\log\left(2^a +1 \right)}{\log(2)}$$
for some $a \in \mathbb{R}$. The authors then claim that if $...
- 961
4
votes
0
answers
341
views
How to evaluate $\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$
I am trying to evaluate
$$\int _0^1\frac{\ln \left(1-x\right)\operatorname{Li}_3\left(x\right)}{1+x}\:dx$$
But I am not sure what to do since integration by parts is not possible here.
I tried using a ...
user809806
1
vote
1
answer
59
views
Further Stirling number series resummation
\begin{equation}
\sum_{m=1}^\infty\sum_{n=1}^\infty (-1)^{n } \frac{S_m^{(3)}}{m! n}(-1 + u)^{(m + n - 1)} (\frac{x}{-1 + x})^m
\end{equation}
Note: $S^{(3)}_m$ belongs to the Stirling number of the ...
- 99
0
votes
0
answers
59
views
Stirling number series resummation
\begin{equation}\sum_{m=1}^{\infty}\frac{a_1^3 S_m^{(3)} (u-1)^{m-1}
\left(\frac{x}{x-1}\right)^m}{m!}\end{equation}
Does somebody know the result of this resummation?
Note:
$S_m^{(3)} $ belongs to ...
- 99
0
votes
0
answers
82
views
General expression of a triangle sequence
\begin{gather*}
\frac{1}{4} \\
\frac{1}{4} \quad \frac{1}{4} \\
\frac{11}{48} \quad \frac{1}{4} \quad \frac{11}{48} \\
\frac{5}{24} \quad \frac{11}{48} \quad \frac{11}{48} \quad \frac{5}{24} \\
\frac{...
- 99
0
votes
1
answer
53
views
General expression of a (maybe 3 or 2 dim) sequence [closed]
$\frac{1}{2}$
$\frac{1}{4}$ $\frac{1}{2}$
$\frac{1}{6}$ $\frac{1}{4}$ $\frac{11}{24}$
$\frac{1}{8}$ $\frac{1}{6}$ $\frac{11}{48}$ $\frac{5}{12}$
$\frac{1}{10}$ $\frac{1}{8}$ $\frac{11}{...
- 99
5
votes
1
answer
223
views
How to evaluate $\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\right)\:dx$
I want to evaluate $$\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\right)\:dx$$
But I've not been successful in doing so, what I tried is
$$\int _0^1\ln \left(\operatorname{Li}_2\left(x\right)\...
user809806
0
votes
0
answers
50
views
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
0
votes
1
answer
97
views
dilogarithm property.
Prove that
$\mathrm{Li}_{2}(-z)+\mathrm{Li}_{2}\left(\frac{z}{1+z}\right)=-\frac{1}{2} \ln ^{2}(1+z)$
I tried to paint in the rows, but I did not succeed. I don't have any more ideas.
- 222
1
vote
1
answer
101
views
Integral $\int_{0}^{e} \frac{\operatorname{W(x)} - x}{\operatorname{W(x)} + x} dx$
$$\int_{0}^{e} \frac{\operatorname{W(x)} - x}{\operatorname{W(x)} + x} dx = 2 \operatorname{Li_2(-e)} - e + \frac{\pi^2}{6} - \log(4) + 4 \log(1 + e)≈-0.819168$$
As usual I prefer to know if there is ...
0
votes
0
answers
61
views
How can we show that the multi-polylogarithmic function $L_{\underbrace{1,\ldots,1}_n}(z)=\frac{1}{n!}(L_1(z))^n$
How can we show that the multi-polylogarithmic function
$$L_{\underbrace{1,\ldots,1}_n}(z)=\frac{1}{n!}(L_1(z))^n.$$
Here $L_1(z)=-log(1-z)$.
I know that $\frac{d}{dz}L_{k_1,\ldots,k_r}(z)=\frac{1}{1-...
- 517