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
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 424
1
vote
1
answer
59
views
Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 812
3
votes
0
answers
121
views
Show that $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$ (polylogarithm)
I am working with the polylogarithm function and want to find closed expressions for $\textrm{Li}_2(e^{ix})$.
If I plot the function $\mathfrak{Re}(\textrm{Li}_2(e^{ix}))$ I get $y=\dfrac{x^2}{4}-\...
- 889
1
vote
1
answer
60
views
Imaginary part of the dilogarithm of an imaginary number
I am wondering if I can simplify
$${\rm Im} \left[ {\rm Li}_2(i x)\right] \ , $$
in terms of more elementary functions, when $x$ is real (in particular, I am interested in $0<x<1$). I checked ...
- 305
3
votes
0
answers
142
views
Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$
Prove that $$-\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
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
98
views
Using Contour Integral to find the value of $\int_{-1}^{+1}\frac{\ln{(1+t)}}{t}dt$
$\newcommand{LogI}{\operatorname{Li}}$
We know that the value of $\LogI_{2}(-1)$ is -$\frac{\pi^2}{12}$ and $\LogI_{2}(1)$ is $\frac{\pi^2}{6}$. The value of the polylogarithms has already been ...
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, ...
- 5,726
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, ...
- 34.2k
2
votes
1
answer
326
views
Evaluate $\Im(\operatorname{Li}_3(2i) + \operatorname{Li}_3(\frac i2))$
Applying the trilogarithm identity
$$ \operatorname{Li}_{3}\left(z\right) - \operatorname{Li}_{3}\left(1 \over z\right) =
-{1 \over 6}\ln^{3}\left(-z\right) -
{\pi^{2} \over 6}\ln\left(-z\right)\tag{1}...
- 99.7k
2
votes
2
answers
226
views
Evaluation of a log-trig integral in terms of the Clausen function (or other functions related to the dilogarithm)
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ via the definite integral
$$\mathcal{I}{\left(a,\theta\right)}:=\int_{0}^{\theta}\mathrm{d}\varphi\,\ln{\left(1-2a\cos{\left(\...
- 30.7k
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 ...
- 8,369
2
votes
1
answer
111
views
Evaluate $\sum_{n\geq1} \frac{(-1)^{n+1}H_n^2}{(n+1)^2}$.
I am looking for a closed for $$\sum_{n\geq1} \frac{(-1)^{n+1}H_n^2}{(n+1)^2}.$$ I believe there is a closed form for the sum as we have seen in [1] which poses as, presumably, a more difficult sum of ...
- 125
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
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
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
$$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$
...
- 158
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 $...
- 6,277
3
votes
0
answers
75
views
Approaching a branch point along different paths
There's a very nice characterization of the three main types of isolated singularities of an analytic function $f(z)$ that takes oriented curves $\gamma$ that terminate at the singularity and ...
- 6,280
3
votes
2
answers
304
views
Jump of dilogarithm
I am reading about the dilogarithm function
$$ \mathrm{Li}_2(z):= - \int_0^z \frac{\log(1-u)}{u}du, \quad z \in \mathbb{C} \backslash [1, \infty).$$
I found it stated that the "jump" of the ...
- 6,326
3
votes
1
answer
500
views
Bose-Einstein function as real part of polylogarithm: $\overline{G}_{s}(x)= \Re \mathrm{Li}_{s+1}(e^x)$
For real $x<0$ the Bose-Einstein integral of order $s$ is given at https://dlmf.nist.gov/25.12.E15 as
$$G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^{t-x}
-1}\mathrm{d}...
- 18.9k
2
votes
1
answer
686
views
Calculate a $\operatorname{Li}_{2}(-1)$ using Integral Representation
$\newcommand{LogI}{\operatorname{Li}}$
I know that $\LogI_{2}(-1)=-\frac{\pi^2}{12}$, but I have never seen a proof of this result without using a functional identity of the Dilogarithm or a series ...
- 436
0
votes
0
answers
135
views
Validity of argument in dilogarithm identities on Wolfram
I've come across a series of identities existing between dilogarithms and powers of logarithms but I am not sure about when such equations are valid in terms of the restriction of the domain of the ...
- 2,850
3
votes
3
answers
449
views
Indefinite integral $\int \arctan^2 x dx$ in terms of the dilogarithm function
I read about the integral
$$\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],$$
as suggested ...
- 7,613
1
vote
1
answer
1k
views
on the (double) discontinuity of dilogarithm along a branch cut
Define the function
$$Li_s(z)=\sum_{k=1}^\infty \frac{z^k}{k^s}$$
for |z|<1. Let's focus on $s=2$.
It can be extended to a holomorphic function on $\mathbb C \setminus [1,\infty)$
$$Li_2(z)= -\...
- 2,390
1
vote
2
answers
640
views
Sum of Dilogarithm and its Complex Conjugate
$$\newcommand{\dl}[1]{\operatorname{Li}_2\left( #1\right)}$$
I have the following expression involving Dilogarithims, where $z\in\mathbb{R}$ and $b\in\mathbb{C}\setminus\lbrace 0\rbrace$:
$$f(z,b) = ...
- 13
3
votes
1
answer
709
views
What's about $\int_0^\infty x^{-z}Li_{z+2}(e^{-xz})dx$ as $-\zeta(3)z^2\Gamma(-z)$?
Because $$-\int_0^\infty \frac{e^{-zt}}{t^z}dt=z^2\Gamma(-z),$$
holds for $0<\Re z<1$ then using the change of variable $t=nx$ one has that $$-\frac{1}{n^{z-1}}\int_0^\infty \frac{e^{-znx}dx}{x^...
user243301
1
vote
0
answers
98
views
What's the worst sequence that still leads to a converging series?
As a background, I'm initially interested in sequences $a_n$ giving rise to functions $\sum_{n=0}^\infty a_nx^n$ for $x\in(0,1)$ and their diverging behavior for $x$ to $1$. E.g. the geometric series $...
- 12.3k
0
votes
1
answer
406
views
Evaluating Fermi Dirac integrals of order j<0
The complete Fermi Dirac integral (I'm purposely leaving off the gamma prefactor)...
$$F_j(x) =\int\limits_{0}^{\infty} \frac{t^j}{e^{t-x}+1} \: dt$$
is generally defined for j>-1. Is there a way to ...
- 1
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