Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
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$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 906
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0
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$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 \operatorname{Li}_{2}\left(\frac12 \...
- 906
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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
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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
$$f(z)=...
- 2,207
3
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Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
- 203
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How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
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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.
$$\sum\limits_{M=2}^{\infty}\sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}\pi \csc(n \pi) \Gamma(M)}{(...
- 906
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$Im[\log(\frac{3}{2})Li_{2}(\frac{3}{2})-Li_{3}(\frac{3}{2})+2Li_{3}(3)]=-\frac{\pi}{2}\log^{2}(2)-\frac{3\pi}{2}\log^{2}(3)+\pi\log(2)\log(3)$
When working on another problem, I got the following expression
\begin{align}
& \Im\left[\log\left(\frac32\right) \operatorname{Li}
_{2} \left(\frac32\right) - \operatorname{Li}
_{3}\left(\frac32\...
- 906
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Is there a closed form solution for the sum $\sum\limits_{M=2}^{\infty} \sum\limits_{n=M}^{\infty} \frac{2^{-M}3^{M-n-1}}{(M-n-1)(n+1)^{2}}$?
While working on another problem, this came up as a sub-step:
$$
\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}}
$$
...
- 906
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How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
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Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
- 5,352
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Simplify the Laplace Transform for $E_{i}(-y)^{2}$
I want to simplify the Laplace transform expression of $E_{i}(-y)^{2}$, where $E_{i}(y)$ is the exponential integral defined by $E_{i}(y) = -\int\limits_{-y}^{\infty} \frac{e^{-t}}{t} dt$.
Question: ...
- 906
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Polylogarithmically solving $\int\frac{\log(a_1x+b_1)\cdots\log(a_nx+b_n)}{px+q}\,dx$
I am now trying a direct approach to solving my question about
$$\int_0^\infty\frac{\arctan a_1x\arctan a_2x\dots\arctan a_nx}{1+x^2}\,dx$$
where the $a_i$ are all positive. Note that the $\arctan$s ...
- 104k
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Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)
Define
$$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\...
- 104k
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How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
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