Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
546
questions
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votes
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+100
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 814
0
votes
0
answers
88
views
$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 \operatorname{Li}_{2}\left(\frac12 \...
- 367
4
votes
0
answers
123
views
Definite integral involving exponential and logarith function
Working with Dilogarimth function, we get the following definite integral
$$\int_0^{\infty}\frac{t^2\,\ln^{n}(t)}{(1-e^{x\,t})(1-e^{y\,t})}\,dt$$
with $n=1,2,3,...$ and $x,y>0$.
I wonder if is ...
- 1,774
3
votes
1
answer
56
views
Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
- 2,022
0
votes
0
answers
34
views
Asymptotic expansions of sums via Cauchy's integral formula and Watson's lemma
I am looking for references that deal with the asymptotic expansions of sums of the form
$$s(n)=\sum_{k=0}^n g(n,k)$$
using the (or similar to) following method.
We have the generating function
$$f(z)=...
- 2,217
6
votes
2
answers
198
views
Computing closed-form of $\int_0^{\infty}\frac{\arctan x}{a^2x^2+1}\,dx$
Find the close form of the integral
$$\int_0^{\infty}\frac{\arctan x}{a^{2}x^2+1}\,dx,\qquad a > 0.$$
I think this integral related with polylogarithm function.
My attempt as follows:
Let $$I(b)=\...
- 90.9k
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
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 90.9k
0
votes
1
answer
38
views
Upper bound of a polylogarithm type series
The series is
$\sum_{n=0}^{\infty}\left(1+\frac{n}{a}\right)^bx^n,$
where $a$ and $b$ are positive real numbers, $x\in[0,1]$.
The sum of the series diverges when $x\to1$. I want to get an upper bound ...
- 267k
3
votes
1
answer
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Is there a closed form solution for the sum $\sum\limits_{M=2}^{\infty} \sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}}{(M-n-1)(n+1)^{2}}$?
While working on another problem, this came up as a sub-step:
$$
\sum_{M\ =\ 2}^{\infty}\,\,\,\sum_{n\ =\ M}^{\infty}\
{2^{-M}\ 3^{M - n - 1} \over \left(M - n - 1\right)\left(n + 1\right)^{\,2}}
$$
...
- 90.9k
4
votes
1
answer
224
views
Closed form for $\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(n+1)^2 \Gamma(M-n)\Gamma(n+1)}$
I encountered this expression generated by mathematica as a sub-step in a problem I am solving.
$$\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(...
- 30.7k
8
votes
1
answer
285
views
Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$
with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$
$$I(a,b)=
\frac\pi4\left(\frac{\pi^2}6
-\Li\...
- 6,277
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votes
1
answer
53
views
$Im[\log(\frac{3}{2})Li_{2}(\frac{3}{2})-Li_{3}(\frac{3}{2})+2Li_{3}(3)]=-\frac{\pi}{2}\log^{2}(2)-\frac{3\pi}{2}\log^{2}(3)+\pi\log(2)\log(3)$
When working on another problem, I got the following expression
\begin{align}
& \Im\left[\log\left(\frac32\right) \operatorname{Li}
_{2} \left(\frac32\right) - \operatorname{Li}
_{3}\left(\frac32\...
- 3,552
14
votes
6
answers
3k
views
Inverse of the polylogarithm
The polylogarithm can be defined using the power series
$$
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
$$
Contiguous polylogs have the ladder operators
$$
\operatorname{Li}_{s+1}(z) ...
- 12.4k
13
votes
6
answers
894
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Strategies for evaluating sums $\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}$
I'm looking for strategies for evaluating the following sums for given $z$ and $m$:
$$
\mathcal{S}_m(z):=\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n},
$$
where $H_n^{(m)}$ is the generalized harmonic ...
- 25.8k
1
vote
0
answers
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How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
