All Questions
Tagged with polylogarithm analysis
8
questions
3
votes
0
answers
119
views
Show that $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$ (polylogarithm)
I am working with the polylogarithm function and want to find closed expressions for $\textrm{Li}_2(e^{ix})$.
If I plot the function $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))$ I get $y=\dfrac{x^2}{4}-\...
- 857
3
votes
0
answers
141
views
Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$
Prove that $$-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$$
where ${\rm Li}_2(x)$ is the Poly Logarithm function and $\zeta(s)$ is the Riemann zeta function
Let $$I=-\int_{0}^{1}...
- 862
4
votes
2
answers
260
views
Logarithmic integral $ \int_0^1 \frac{x}{x^2+1} \, \log(x)\log(x+1) \, {\rm d}x $
At various places e.g.
Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$
and
How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$
logarithmic integrals are connected ...
- 6,277
0
votes
1
answer
57
views
Taylor expansion of $x/W(x)$
For $x>e$, express $x/W(x)$ with respect to $\ln(x)$ and $\ln \ln(x)$, where $W(\cdot)$ is the Lambert-W function.
In Wikipedia, we can find the expression of $W(x)$ with respect to $\ln(x)$ and $...
- 2,760
25
votes
4
answers
1k
views
Evaluate $\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta $
Evaluate
$$\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\theta $$
Several days ago,I found this interesting integral from a paper about generalized log-sine ...
- 5,281
4
votes
1
answer
112
views
Equality with dilogarithms
During some calculations with definite integrals I happened to get the equality
\begin{eqnarray}
2\, \textrm{Li}_2(-\frac{1}{2}) - 2 \, \textrm{Li}_2(\frac{1}{4})+ 2\, \textrm{Li}_2(\frac{2}{3})=
3 \...
- 54.7k
3
votes
1
answer
189
views
Series for reciprocal of polylogarithm?
The series representation of a polylogarithm of order $s$ is given by
$$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$
Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
- 4,048
22
votes
4
answers
1k
views
Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?
Ramanujan gave the following identities for the Dilogarithm function:
$$
\begin{align*}
\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) &=\frac{{...
- 11.6k