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Tagged with polylogarithm integration
271
questions
4
votes
0
answers
112
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Calculate an integral involving polylog functions
Im my recent answer https://math.stackexchange.com/a/4777055/198592 I found numerically that the following integral has a very simple result
$$i = \int_0^1 \frac{\text{Li}_2\left(\frac{i\; t}{\sqrt{1-...
- 12.7k
8
votes
3
answers
1k
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Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$
Is it possible to show
$$
\int_{0}^{1}\frac{K(k)\ln\left[\tfrac{\left ( 1+k \right)^3}{1-k} \right] }{k}
\text{d}k=\frac{\pi^3}{4}\;\;?
$$
where $K(k)$ is the complete elliptic integral of the first ...
- 5,370
1
vote
0
answers
69
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Polylogarithm further generalized
Here I proposed a generalized formula for the polylogarithm. However, because of a slight mistake towards the end, visible prior to the edit, I was unaware that it yields just a result of an integral ...
- 1,037
5
votes
1
answer
193
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Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
2
votes
0
answers
85
views
Complex polylogarithm/Clausen function/Fourier series
Sorry for the confusing title but I'm having a problem and I can phrase the question in multiple different ways.
I was calculating with WolframAlpha
$$\int \text{atanh}(\cos(x))\mathrm{d}x= i \text{Li}...
2
votes
0
answers
68
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Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms
As per the title, I evaluated
$$\int\frac{\log(x+a)}{x}\,dx$$
And wanted to make sure my solution is correct, and if not, where I went wrong in my process. Here is my work.
$$\int\frac{\log(x+a)}{x}\,...
- 1,123
6
votes
2
answers
326
views
How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$
I am trying to prove that
$$\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G,$$
where $G$ is the Catalan constant and $\operatorname{Li}_2(x)$ is the dilogarithm ...
- 25.8k
5
votes
1
answer
288
views
Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $
(This is related to this question).
How would one find the closed forms the integral
$$ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x?
$$
I tried using Nielsen Generalized Polylogarithm as mentioned ...
- 107
1
vote
1
answer
122
views
Show that $\int_0^1 \frac{Li_{1 - 2m}(1 - 1/x)}{x} dx = 0$.
I would like to show that,for $m \geq 2$,
$$I_m := \int_0^1 \frac{\operatorname{Li}_{1 - 2m}(1 - 1/x)}{x} dx = 0$$ where $\operatorname{Li}_{1 - 2m}$ is the $1-2m$ polylogarithm (https://en.wikipedia....
- 2,073
2
votes
1
answer
259
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Generalized formula for the polylogarithm
Some time ago, I discovered the formula for repeated application of $z\frac{d}{dz}$ here. Recently, I thought about taking the function to which this would be applied to be the integral representation ...
- 1,037
5
votes
3
answers
258
views
How to find the exact value of $\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{n^2 \cdot 2^{\frac{n}{2}}} $?
Once I met the identity
$$
\boxed{S_0=\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{2^{\frac{n}{2}}}=1},
$$
I first tried to prove it by $e^{xi}=\cos x+i\sin x$.
$$
\begin{aligned}
\...
- 22.3k
2
votes
1
answer
174
views
Calculate the integral of the given polylogarithm function? $\int_0^1\frac{\operatorname{Li}_ 4(x)}{1+x}dx=?$ [closed]
$$\int_0^1 \frac{\operatorname{Li}_2(-x)\operatorname{Li}_2(x)}{x}\,\mathrm dx=?$$
where $$\operatorname{Li}_2(-x)=\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$$ for $$|x|>1$$
actually my goal is to edit ...
- 71
3
votes
1
answer
244
views
How to solve $\int\frac{x\arctan x}{x^4+1}dx$ in a practical way
I need to evaluate the following indefinite integral for some other definite integral
$$\int\frac{x\arctan x}{x^4+1}dx$$
I found that
$$\int_o^\infty\arctan{(e^{-x})}\arctan{(e^{-2x})}dx=\frac{\pi G}{...
- 2,370
21
votes
3
answers
2k
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Evaluate $\int_0^1\arcsin^2(\frac{\sqrt{-x}}{2}) (\log^3 x) (\frac{8}{1+x}+\frac{1}{x}) \, dx$
Here is an interesting integral, which is equivalent to the title
$$\tag{1}\int_0^1 \log ^2\left(\sqrt{\frac{x}{4}+1}-\sqrt{\frac{x}{4}}\right) (\log ^3x) \left(\frac{8}{1+x}+\frac{1}{x}\right) \, dx =...
- 19.1k
1
vote
0
answers
54
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Dilogarithm Function on Negative Domain
I'm not that good with math, but somehow ended up solving for
$ \int { \ln { (\cosh x) } } \cdot dx $. This has led me to the answer described here. In my case, I need a solution for x > 1, ...
- 11