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3
questions
11
votes
0
answers
255
views
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
$$\int_0^...
- 1,675
3
votes
1
answer
148
views
Explicit value of $\operatorname{Li}_2(1/2-{\rm i}/2)$
When you ask Wolfram Alpha about the value of $\operatorname{Li}_{2}\left(1/2-{\rm i}/2\right)$ it gives you $$
\frac{5\pi^{2}}{96} -
\frac{\ln^{2}\left(2\right)}{8} +
{\rm i}\left[\frac{\pi\ln\left(2\...
2
votes
1
answer
217
views
Can this formula for $\zeta(3)$ be proven or simplified further?
This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
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