All Questions
Tagged with polylogarithm closed-form
125
questions
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 ...
- 203
1
vote
0
answers
64
views
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+...
1
vote
0
answers
50
views
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
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 $\newcommand{Li}{\operatorname{Li}_2}$
$$I(a,b)=
\frac\pi4\left(\frac{\pi^2}6
-\Li\...
- 104k
12
votes
2
answers
499
views
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
- 3,871
3
votes
0
answers
186
views
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$?
here in my answer I got real part for polylogarithm function at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\...
- 1,577
4
votes
0
answers
111
views
Calculate an integral involving polylog functions
Im my recent answer https://math.stackexchange.com/a/4777055/198592 I found numerically that the following integral has a very simple result
$$i = \int_0^1 \frac{\text{Li}_2\left(\frac{i\; t}{\sqrt{1-...
- 12.7k
4
votes
0
answers
83
views
Closed form of dilogarithm fucntion involving many arctangents
I am trying to find closed form for this expression:
$$ - 2{\text{L}}{{\text{i}}_2}\left( {\frac{1}{3}} \right) - {\text{L}}{{\text{i}}_2}\left( {\frac{1}{6}\left( {1 + i\sqrt 2 } \right)} \right) - {\...
- 2,702
5
votes
1
answer
193
views
Evaluating $\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{(ax)}\operatorname{arsinh}{(bx)}}{x}$ in terms of polylogarithms
Define the function $\mathcal{I}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by the definite integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{1}\mathrm{d}x\,\frac{\operatorname{arsinh}{\left(ax\right)}\...
- 30.7k
1
vote
1
answer
60
views
Imaginary part of the dilogarithm of an imaginary number
I am wondering if I can simplify
$${\rm Im} \left[ {\rm Li}_2(i x)\right] \ , $$
in terms of more elementary functions, when $x$ is real (in particular, I am interested in $0<x<1$). I checked ...
- 305
5
votes
1
answer
286
views
Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $
(This is related to this question).
How would one find the closed forms the integral
$$ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x?
$$
I tried using Nielsen Generalized Polylogarithm as mentioned ...
- 105
2
votes
2
answers
93
views
Finding a recurrence relation to evaluate $\int_{a}^{1}\mathrm{d}x\,\frac{x^{n}}{\sqrt{1-x^{2}}}\ln{\left(\frac{x+a}{x-a}\right)}$
For each $n\in\mathbb{Z}_{\ge0}$, define the function $\mathcal{J}_{n}:(0,1)\rightarrow\mathbb{R}$ via the doubly improper integral
$$\mathcal{J}_{n}{\left(a\right)}:=\int_{a}^{1}\mathrm{d}x\,\frac{x^{...
- 30.7k
3
votes
2
answers
412
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 ...
- 51
4
votes
1
answer
257
views
Find closed-form of: $\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$
I found this integral: $$\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$$
And it seems look like this problem but i don't know how to process with this one.
First, i tried to use series of $\frac{x}{x^...
- 2,702
3
votes
1
answer
215
views
Is there an analytic solution to $\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$?
Is there a sensible analytic solution to the following integral: $$\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$$ where all constants are real and $C>0$.
This integral is part of the third term ...
- 31