All Questions
11
questions
1
vote
0
answers
231
views
Improper Integral Involving Hyperbolic Cotangent
I am trying to evaluate the following integral:
$$\int_0^\infty \frac{\frac{1}{x}-\pi\coth(\pi x)}{x^2+4}dx$$
I'm not sure if a closed-form exists, so far I only know the decimal approximation to be $\...
- 63
6
votes
3
answers
409
views
Evaluate $\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$
I'm trying to evaluate the following integral:
$$\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$$
I was playing around trying to numerically approximate the answer with known constants and found ...
- 2,324
46
votes
10
answers
8k
views
Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$
It appears that
$$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$
(so far I have about $1000$ decimal digits to confirm that).
After changing variable $...
- 47.3k
11
votes
2
answers
311
views
Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$
How could we prove that
$${\LARGE\prod_{\Large n\ge0}}~\frac{\cosh\left(n^2+n+\dfrac12\right)+i\sinh\left(n+\dfrac12\right)}{\cosh\left(n^2+n+\dfrac12\right)-i\sinh\left(n+\dfrac12\right)}~=~i$$
...
- 48.5k
16
votes
3
answers
1k
views
Closed form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$
While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms:
$$
I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} \frac{\...
- 12.4k
5
votes
2
answers
364
views
Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$
Reading in some tables pages I found
$$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$
I try to split in two sum using the roots of the $\tanh$...
user167276
35
votes
4
answers
2k
views
Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$
How to prove the following conjectured identity?
$$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$
It holds ...
- 47.3k
16
votes
2
answers
519
views
Closed form for ${\large\int}_0^\infty\frac{x\,\sqrt{e^x-1}}{1-2\cosh x}\,dx$
I was able to calculate
$$\int_0^\infty\frac{\sqrt{e^x-1}}{1-2\cosh x}\,dx=-\frac\pi{\sqrt3}.$$
It turns out the integrand even has an elementary antiderivative (see here).
Now I'm interested in a ...
- 13.2k
11
votes
1
answer
567
views
A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$
I need to find a closed form for this integral:
$$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$
A numerical integration results in an approximation $\...
- 47.3k
30
votes
5
answers
1k
views
Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
Let
$$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$
where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant.
Here are values of $f(a)$ at some ...
- 47.3k
37
votes
2
answers
929
views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$
Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
- 13.2k