All Questions
Tagged with polylogarithm complex-analysis
41
questions
2
votes
0
answers
40
views
How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 91k
1
vote
1
answer
59
views
Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 4,249
3
votes
0
answers
121
views
Show that $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$ (polylogarithm)
I am working with the polylogarithm function and want to find closed expressions for $\textrm{Li}_2(e^{ix})$.
If I plot the function $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))$ I get $y=\dfrac{x^2}{4}-\...
- 3,940
1
vote
1
answer
60
views
Imaginary part of the dilogarithm of an imaginary number
I am wondering if I can simplify
$${\rm Im} \left[ {\rm Li}_2(i x)\right] \ , $$
in terms of more elementary functions, when $x$ is real (in particular, I am interested in $0<x<1$). I checked ...
- 268k
3
votes
0
answers
142
views
Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$
Prove that $$-\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}...
- 910
3
votes
2
answers
416
views
Imaginary part of dilogarithm
I have evaluated a certain real-valued, finite integral with no general elementary solution, but which I have been able to prove equals the imaginary part of some dilogarithms and can write in the ...
- 61
2
votes
1
answer
311
views
Real Part of the Dilogarithm
It is well known that
$$\frac{x-\pi}{2}=-\sum_{k\geq 1}\frac{\sin{kx}}{k}\forall x\in(0,\tau),$$
which gives
$$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$
...
- 2,019
2
votes
2
answers
797
views
Branch Points of the Polylog function
The polylogarithm
$$
{\rm Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s}
$$
has obvious branch points at $z=1$.
For integers $s\leq 0$ it is a rational function with a pole of order $1-s$ at $z=1$. If $...
- 42.7k
17
votes
2
answers
894
views
A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ...
Question: How to show the relation
$$
J:=\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt
=-\frac 1{64}\pi^4
$$
(using a "minimal industry" of relations, ...
- 5,495
2
votes
2
answers
288
views
Does the dilogarithm function (which is multi-valued) have a single-valued inverse?
The $p$-logarithm is defined for $|z|<1$ by
$$\text{Li}_p(z)=\sum_{n=1}^\infty\frac{z^n}{n^p}$$
and defined elsewhere in $\mathbb C$ by analytic continuation, though it may be multi-valued, ...
- 597
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}
...
- 1,521
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
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
$$
Li_s(s)=\sum_{n=1}^{\infty} \frac{z^...
CommunityBot
- 1
5
votes
3
answers
320
views
Is there a closed-form for $\sum_{n=0}^{\infty}\frac{n}{n^3+1}$?
I'm reading a book on complex variables (The Theory of Functions of a Complex Variable, Thorn 1953) and the following is shown:
Let $f(z)$ be holomorphic and single valued in $\mathbb{C}$ except at a ...
- 113k
3
votes
3
answers
449
views
Indefinite integral $\int \arctan^2 x dx$ in terms of the dilogarithm function
I read about the integral
$$\int \arctan^2 x dx$$ in this old post: Evaluation of $\int (\arctan x)^2 dx$
By replacing
$$\arctan x = -\frac{i}{2}\left[\log(1+ix) - \log(i-ix)\right],$$
as suggested ...
- 5,793