All Questions
Tagged with polylogarithm complex-analysis
41
questions
10
votes
2
answers
509
views
Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$
I experimentally discovered the following conjectures:
$$\Re\Big[1800\operatorname{Li}_3(i\,\phi)-24\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=100\ln^3\phi-47\,\pi^2\ln\phi-...
- 47.3k
5
votes
1
answer
753
views
On what domain is the dilogarithm analytic?
The series $\displaystyle\sum \dfrac{z^n}{n^2}$ converges for $\lvert z\rvert<1$ by the ratio test, meaning that the dilogarithm function $\text{Li}_2(z),$ which is equal to the series $\...
- 16.9k
4
votes
0
answers
301
views
Simplification of an expression involving the dilogarithm with complex argument
Do you think there is a way to get a nice form of the expression below
$$\Im{\left( \text{Li}_2\left(\frac{3}{5}+\frac{4 i}{5}\right)- \text{Li}_2\left(-\frac{3}{5}+\frac{4 i}{5}\right)+ \text{Li}_2\...
- 44.4k
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:
$$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
- 47.3k
8
votes
2
answers
2k
views
Find the derivative of a polylogarithm function
I was trying to find to which function the next series converges.
$$
\sum_{n=1}^{\infty} \ln(n)z^n
$$
If we take the polylogarithm function $Li_s(z)$ defined as
$$
Li_s(s)=\sum_{n=1}^{\infty} \frac{z^...
- 6,589
4
votes
3
answers
170
views
There's a small detail in this proof on why $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I can't figure out
http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf
Here is a link to the article I have been reading. It's really interesting and easy to follow. What bothers me is a result ...
- 987
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.
$$\operatorname{Re}\big[\operatorname{Li}_2\left(i\right)\big]=-\frac{\pi^2}{48},\hspace{1em}\...
- 1,928
4
votes
1
answer
347
views
On a property of polylogarithm
I have an observation, and I don't know that the following statement is true or not. If not give a counterexample, if it is true prove it, or give a reference about it.
Let $n \in \mathbb{R}$, $z \in ...
- 12.4k
1
vote
1
answer
121
views
Inverse of Higher logarithms
Th polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the ...
- 6,245
16
votes
5
answers
1k
views
Double Euler sum $ \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} $
I proved the following result
$$\displaystyle \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} =- \frac{97}{12} \zeta(6)+\frac{7}{4}\zeta(4)\zeta(2) + \frac{5}{2}\zeta(3)^2+\frac{2}{3}\zeta(2)^3$$
After ...
- 14.4k
15
votes
1
answer
228
views
Simplification of a trilogarithm of a complex argument
Is it possible to simplify the following expression?
$$\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