All Questions
8
questions
1
vote
0
answers
128
views
Conjectured closed form for ${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt { 2}}{2}} \right) \right)$
With Maple i find this closed form:
${\it {Li_2}} \left( 1-{\frac {\sqrt {2}}{2}}-i \left( 1-{\frac {\sqrt {
2}}{2}} \right) \right)$=$-{\frac {{\pi}^{2}}{64}}-{\frac { \left( \ln \left( 1+\sqrt {2}
...
- 303
1
vote
0
answers
145
views
Dilogarithm of a negative real number outside unit circle
The dilogarithm is defined in $\mathbb{C}$ as
$$
Li_2(z) = -\int_0^1 \frac{\ln(1 - zt)}{t} dt
$$
Because $1-zt \in \mathbb{C}$, then you can write $\ln(1 - zt) = \ln|1 - zt| + i·\arg(1 - zt)$
As ...
- 539
0
votes
0
answers
135
views
Validity of argument in dilogarithm identities on Wolfram
I've come across a series of identities existing between dilogarithms and powers of logarithms but I am not sure about when such equations are valid in terms of the restriction of the domain of the ...
- 2,850
5
votes
2
answers
438
views
Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$
Is there a closed form (free of polylogs and ...
- 369
20
votes
3
answers
908
views
Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$
I numerically discovered the following conjecture:
$$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
- 47.3k
22
votes
2
answers
3k
views
Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$$\operatorname{Re}\big[\operatorname{Li}_2\left(i\right)\big]=-\frac{\pi^2}{48},\hspace{1em}\...
- 1,928
4
votes
1
answer
347
views
On a property of polylogarithm
I have an observation, and I don't know that the following statement is true or not. If not give a counterexample, if it is true prove it, or give a reference about it.
Let $n \in \mathbb{R}$, $z \in ...
- 12.4k
15
votes
1
answer
228
views
Simplification of a trilogarithm of a complex argument
Is it possible to simplify the following expression?
$$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$
where
$$\...
- 3,320