All Questions
3
questions
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
6
votes
0
answers
362
views
Evaluate two integrals involving $\operatorname{Li}_3,\operatorname{Li}_4$
I need to evaluate
$$\int_{1}^{\infty}
\frac{\displaystyle{\operatorname{Re}\left (
\operatorname{Li}_3\left ( \frac{1+x}{2} \right ) \right )
\ln^2\left ( \frac{1+x}{2} \right ) }}{x(1+x^2)} \...
- 5,370
12
votes
3
answers
460
views
How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$
I am trying evaluating this
$$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