All Questions
Tagged with polylogarithm contour-integration
9
questions
1
vote
0
answers
80
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Simplify the Laplace Transform for $E_{i}(-y)^{2}$
I want to simplify the Laplace transform expression of $E_{i}(-y)^{2}$, where $E_{i}(y)$ is the exponential integral defined by $E_{i}(y) = -\int\limits_{-y}^{\infty} \frac{e^{-t}}{t} dt$.
Question: ...
- 814
4
votes
3
answers
136
views
I need help evaluating the integral $\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$
I was playing around with the integral: $$\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$$
I couldn't find a way of solving it, but I used WolframAlpha to find that the integral evaluated ...
- 71
1
vote
0
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59
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Contour integration with polylogarithm
Starting from Bose-Einstein integral representation of the polylogarithm
$$Li_{s}(z) = \frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t/z - 1}dt \quad \quad(1)$$
it's not too hard to obtain the ...
- 342
6
votes
0
answers
362
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Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 5,362
12
votes
3
answers
460
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How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$
I am trying evaluating this
$$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}.$$
Maybe $J(k)$ ...
- 5,362
1
vote
0
answers
98
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Using Contour Integral to find the value of $\int_{-1}^{+1}\frac{\ln{(1+t)}}{t}dt$
$\newcommand{LogI}{\operatorname{Li}}$
We know that the value of $\LogI_{2}(-1)$ is -$\frac{\pi^2}{12}$ and $\LogI_{2}(1)$ is $\frac{\pi^2}{6}$. The value of the polylogarithms has already been ...
5
votes
3
answers
320
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Is there a closed-form for $\sum_{n=0}^{\infty}\frac{n}{n^3+1}$?
I'm reading a book on complex variables (The Theory of Functions of a Complex Variable, Thorn 1953) and the following is shown:
Let $f(z)$ be holomorphic and single valued in $\mathbb{C}$ except at a ...
- 8,389
0
votes
0
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173
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Contour Integral involving Dilogarithmic functions
I am considering the contour integral:
$\int Li_2\left( \frac{1-z}{2}\right)Li_2\left( \frac{z-1}{2z}\right) \frac{dz}{z}$.
The contour of integration is the unit circle excluding the pole $z = 0$. $...
- 407
17
votes
2
answers
834
views
Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$
I would like to seek your assistance in computing the sum
$$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$
I am stumped by this sum. I have tried summing the residues of $\displaystyle f(z)=\frac{\pi\...
- 5,604