Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
114
questions with no upvoted or accepted answers
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Are these generalizations known in the literature?
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
- 25.8k
11
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Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$
I tried to solve this integral and got it, I showed firstly
$$\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^...
- 1,577
11
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Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?
Using
$$\text{Li}_{2a+1}(x)-\text{Li}_{2a+1}(1/x)=\frac{i\,\pi\ln^{2a}(x)}{(2a)!}+2\sum_{k=0}^a \frac{\zeta(2a-2k)}{(2k+1)!}\ln^{2k+1}(x)\tag{1}$$
and
$$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
- 25.8k
9
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answers
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Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
- 5,362
8
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Powerful Integral $\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt$
This integral can be found in Cornel's book, (Almost) Impossible Integral, Sums and Series page $97$ where he showed that
$$\int_0^x\frac{t\ln(1-t)}{1+t^2}\ dt=\frac14\left(\frac12\ln^2(1+x^2)-2\...
- 25.8k
8
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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,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
8
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Why does the tribonacci constant have a trilogarithm ladder?
When I came across the dilogarithm ladders of Coxeter and Landen, namely,
$$\text{Li}_2\Big(\frac1{\phi^6}\Big)-4\text{Li}_2\Big(\frac1{\phi^3}\Big)-3\text{Li}_2\Big(\frac1{\phi^2}\Big)+6\text{Li}_2\...
- 54.9k
6
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1
answer
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Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$
this integral got posted on a mathematics group by a friend
$$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$
I tried seeing what I'd get from ...
- 231
6
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362
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Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 5,362
6
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305
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Does there exist a closed form for $\int_0^{\pi/2}\frac{x^2\ \text{Li}_2(\sin^2x)}{\sin x}dx$?
I am not sure if there exists a closed form for
$$I=\int_0^{\pi/2}\frac{x^2\ \text{Li}_2(\sin^2x)}{\sin x}dx$$
which seems non-trivial.
I used the reflection and landen's identity, didn't help much.
...
- 25.8k
6
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0
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182
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Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}8)$?
This was inspired by Oksana's post. Let, $$a = \ln 2 \quad\quad\\ b = \ln 3\quad\quad\\ c = \ln 5\quad\quad$$
then the following,
\begin{align}
A &= \text{Li}_3\left(\frac12\right)\\
B &= \...
- 54.9k
6
votes
0
answers
197
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Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$
I am trying to evaluate the following integral
$$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0<...
- 93
5
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answers
281
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Is there a closed form without MZV for $ \sum _{k=1}^{\infty }\frac{H_k}{k^6\:2^k}$?
While evaluating the weight $7$ integral $\displaystyle \int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\right)}{1+x}\:dx$
I managed to prove that
$$\int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\...
- 2,646
5
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0
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161
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Verifying closed form evaluation of an Ising-class multiple integral
For $n\in\mathbb{N}\land n\ge2$, define the so-called Ising-class integral of the third kind, $E_{n}$, via the sequence of $\left(n-1\right)$-dimensional integrals
$$E_{n}:=2\int_{\left[0,1\right]^{...
- 30.7k
4
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Definite integral involving exponential and logarith function
Working with Dilogarimth function, we get the following definite integral
$$\int_0^{\infty}\frac{t^2\,\ln^{n}(t)}{(1-e^{x\,t})(1-e^{y\,t})}\,dt$$
with $n=1,2,3,...$ and $x,y>0$.
I wonder if is ...
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