All Questions
21
questions
1
vote
0
answers
109
views
Evaluate $\int_{0}^{\infty }\!{\frac {t}{{{\rm e}^{3\,\pi\,t}}-1}\ln \left( {t} ^{2}+{\frac{1}{4}} \right) }\,{\rm d}t$
I continue my work about the integral as $\int_{0}^{\infty }\!{\frac {t\ln \left( {t}^{2}+{z}^{2} \right) }{{
{\rm e}^{a\pi\,t}}-1}}\,{\rm d}t$.
Recently, i find the general formula for $\int_{0}^{\...
- 303
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
3
votes
1
answer
326
views
Closed form solution for $\int_0^{\infty } \frac{\sin ({n}/{x})}{e^{2 \pi x}-1} \, dx$
Is there a closed form solution for the following integral
$$\int_0^{\infty } \frac{\sin \left(\frac{n}{x}\right)}{e^{2 \pi x}-1} \, dx$$
for $n>0$
- 4,492
4
votes
3
answers
182
views
What is the exact value of $\int^\infty_0\frac{\sin^2 x}{x^{5/2}}\,dx$
I would like to get the exact value of the following integral.
$$\int^\infty_0\frac{\sin^2 x}{x^{5/2}}\,dx$$
I was able to prove the convergence as well. But I don't how to compute its exact value....
- 24.2k
4
votes
1
answer
155
views
Finding a closed form for $\int_0^{\infty} \frac{\sin(x/\epsilon)}{1+x^2}dx$ in terms of $\epsilon$?
$$\int_0^{\infty} \frac{\sin(x/\epsilon)}{1+x^2}dx$$
We can use complex analysis to show that $\int_0^{\infty} \frac{\cos(x/\epsilon)}{1+x^2}dx = \frac{\pi}{2}e^{-1/\epsilon}$ but this sin version is ...
- 51
5
votes
2
answers
461
views
Write $\sum\limits_{n=0}^\infty e^{-xn^3}$ in the form $\sum\limits_{n=-\infty}^\infty a_nx^n$
This is a very simple question; I apologize if it has already been asked here. Define the following function (superficially similar to a theta function):
$$\varsigma(x)=\sum_{n=1}^\infty e^{-xn^3}$$
...
- 3,343
8
votes
2
answers
173
views
How to prove $\sum_{k=1}^\infty\frac{k^k}{k!}x^k=\frac{1}{2}$ where $x=\frac{1}{3}e^{-1/3}$
How to prove that
$$
\sum_{k=1}^\infty\frac{k^k}{k!}x^k=\frac{1}{2}, ~\text{where}~~ x=\frac{1}{3}e^{-1/3}~?
$$
I found this sum in my notes, but I don't remember where I got it. Any hints or ...
- 1,695
9
votes
0
answers
387
views
Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$
Is it possible to calculate the sum
$$
\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}
$$
in closed form?
Formal naive argument gives
$$
\sum _{n=0}^{\infty} \...
- 1,695
2
votes
3
answers
246
views
Closed form for $\int_0^1 \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right) ~d u$
The parameter $\lambda$ is complex and it's not on the real axis.
There are some similar cases:
Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$
Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $
...
- 247
2
votes
1
answer
122
views
How to find Real Part of PolyLog[3,(1-i)] in closed form
$ \Re \bigg(\text{Li}_3(1-i)\bigg)=\frac{\pi^a}{b}\ln(2)+\frac{c}{d}\zeta(e)$ has an approximate value of .8711588834109380
if $a=1 , b=-3415 , c=34 , d=39 , e=19$ are substituted into the closed ...
- 750
5
votes
2
answers
438
views
Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$
Is there a closed form (free of polylogs and ...
- 369
20
votes
3
answers
908
views
Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$
I numerically discovered the following conjecture:
$$\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)\stackrel{\color{gray}?}=\frac{7\pi^2}{48}-\frac{\arctan^22}3-\frac{\arctan^23}6-\frac18\ln^2\!...
- 47.3k
5
votes
1
answer
156
views
How to integrate $\int_0^{\infty}\frac{e^{-(t+\frac1t)}}{\sqrt t} dt$?
This is a problem given in my homework . I have to find the integral$$\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t}dt$$
I am trying to use integral representation of the gamma ...
- 6,198
48
votes
1
answer
2k
views
Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $
I need the method to evaluate this integral (the closed-form if possible).
$$
\int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx
$$
I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
- 23.4k
22
votes
2
answers
3k
views
Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$
We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g.
$$\operatorname{Re}\big[\operatorname{Li}_2\left(i\right)\big]=-\frac{\pi^2}{48},\hspace{1em}\...
- 1,928