All Questions
Tagged with polylogarithm closed-form
125
questions
8
votes
0
answers
413
views
More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$
I. In this post, the OP asks about the particular log sine integral,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
7
votes
3
answers
615
views
Closed-forms for the integral $\int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$?
(This is related to this question.)
Define the integral,
$$I_n = \int_0^1\frac{\operatorname{Li}_n(x)}{1+x}dx$$
with polylogarithm $\operatorname{Li}_n(x)$. Given the Nielsen generalized polylogarithm ...
- 54.9k
5
votes
4
answers
394
views
Integral: $\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$
I am trying to evaluate $$P=\frac\pi2\sum_{n\geq1}\frac{{2n\choose n}}{4^n n^2}$$
I used the beta function to show that
$$P=\int_0^1\frac{\mathrm{Li}_2(x^2)}{\sqrt{1-x^2}}dx$$
IBP:
$$P=\sin^{-1}(x)\...
- 17.3k
5
votes
5
answers
313
views
How can I compute this integral in closed form : $\int_0^{\frac{π}{4}}\ln^2(\tan x)dx$
How can I compute this integral in closed form :
$$\displaystyle\int_{0}^{\displaystyle \tfrac{π}{4}}\ln^{2}\left(\tan x\right)dx$$
How can use Fourier series here ?
$$-2\displaystyle \sum_{n=0}^{\...
user670794
-1
votes
2
answers
103
views
Compute in closed form : $\int_0^{\frac{π}{4}} x\ln(\tan x)\left(1-\frac{1}{\cos^2 x}\right)dx$
Question :
Compute in closed form without use series
$I =\displaystyle\int_0^{\pi / 4} x\ln\left(\tan x\right)\left(1-\frac{1}{\cos^2 x}\right)\,dx$
I think use : $y=\tan x$ then $dy=\frac{1}{\cos^...
user668815
3
votes
5
answers
290
views
Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$
I am trying to find closed form for this integral:
$$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$
Where $a>0$.
My try: Let: $$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$
Then:
$$\...
user668815
6
votes
2
answers
445
views
Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$
Compute the following integration without harmonic series or Fourier series :
$I=\displaystyle\int_0^{\frac{π}{4}}x\ln(\tan x)dx$
Wolfram alpha give $I=\frac{7\zeta(3)-4πC}{16}$
Where $C$ : Catalan'...
user668815
8
votes
1
answer
216
views
closed form for $\int_0^1\frac{\mathrm{Li}_s(x-x^2)}{x-x^2}\mathrm dx$
I am trying to evaluate
$$F(s)=\sum_{n\geq1}\frac1{n^{s+1}{2n\choose n}}$$
I started off by noting that $$\frac1{n{2n\choose n}}=\frac12\int_0^1\left[x-x^2\right]^{n-1}\mathrm dx$$
So
$$F(s)=\int_0^1\...
- 17.3k
16
votes
2
answers
1k
views
Evaluate$\int\limits_0^1 [\log(x)\log(1-x)+\operatorname{Li}_2(x)]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]\mathrm dx$
The following is from Mathematical Analysis $-$ A collection of Problems by Tolaso J. Kos $($Page $27$, Problem $282$$)$
$$\mathfrak{I}=\int\limits_0^1 \left[\log(x)\log(1-x)+\operatorname{Li}_2(x)\...
- 16.2k
5
votes
1
answer
171
views
Are there any non-trivial special values of $\operatorname{Li}_4(z)$?
Denote $\operatorname{Li}_4(z)$ the analytic continuation of $\sum_{n=1}^\infty\frac{z^n}{n^4}$. $z$ is a algebraic number with $|z|\ne 0,1$. Does $\Re\operatorname{Li}_4(z)$ or $\Im\operatorname{Li}...
- 8,679
13
votes
3
answers
702
views
Prove that $\int_0^1\frac{\operatorname{Li}_3(1-z)}{\sqrt{z(1-z)}}\mathrm dz=-\frac{\pi^3}{3}\log 2+\frac{4\pi}3\log^3 2+2\pi\zeta(3)$
While going through the recent questions concerning tagged polylogarithms I stumbled upon this post which asks for a concrete evaluating of a polylogarithmic integral. However the post also states the ...
- 16.2k
9
votes
6
answers
881
views
Show that $\int_0^{\infty}\frac{\operatorname{Li}_s(-x)}{x^{\alpha+1}}\mathrm dx=-\frac1{\alpha^s}\frac{\pi}{\sin(\pi \alpha)}$
I have come across the following integral while going over this list (Problem $35$)
$$\int_0^{\infty}\frac{\operatorname{Li}_s(-x)}{x^{\alpha+1}}\mathrm dx=-\frac1{\alpha^s}\frac{\pi}{\sin(\pi \alpha)...
- 16.2k
7
votes
1
answer
307
views
Integral $\int_0^1 \frac{\log^4(x)}{(1-x)^4}dx$
I am trying to evaluate $$\int_0^1 \frac{\log^4(x)}{(1-x)^4}\,dx$$
This type of integral already has two answers here: Closed form for $\int_0^1 \frac {\log^n(x)}{(1-x)^m} dx$. However I desire to ...
- 28.1k
4
votes
5
answers
805
views
Closed form for ${\large\int}_0^1\frac{\ln^4(1+x)\ln x}x \, dx$
Can someone compute
$$ \int_0^1\frac{\ln^4(1+x)\ln x}x \,dx$$
in closed form?
I conjecture that the answer can be expressed as a polynomial function with rational coefficients in constants of the ...
- 1,707
2
votes
1
answer
229
views
Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram Alpha
I was thinking about the shape of integrals related with $\zeta(3)$ and Catalan's constant, I am saying those in section 3.1 of this Wikipedia. I was thinking in moments of higher order $x^k$ in the ...
user243301