All Questions
93
questions
7
votes
1
answer
171
views
Least number of circles required to cover a continuous function on a closed interval.
Now asked on MO here.
This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to ...
- 6,620
4
votes
2
answers
238
views
Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $
Just for curiosity I want to generalise the Pi function i.e $f(x+1) = f(x)g(x+1)$ for some differentiable function, I know this function probably has no closed form for general functions $g$ as I ...
- 6,620
2
votes
0
answers
70
views
Closed form for ${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right)$
I am trying to find the closed form the expression $${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right).$$ I was able to convert the expression into the series $${...
6
votes
3
answers
399
views
Closed form for $\int_0^1\frac{(x^3-3x^2+x)\log(x-x^2)}{(x^2-x+1)^3}\mathrm dx$
I am looking for a closed form for
$$I=\int_0^1\frac{(x^3-3x^2+x)\log(x-x^2)}{(x^2-x+1)^3}\mathrm dx\approx 0.851035604949$$
Wolfram does not evaluate $I$.
I suspect $I$ has a closed form, because if $...
- 25.7k
5
votes
2
answers
211
views
Prove $_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)=\frac{27}{16}\left(\pi^2-7\zeta(3)\right) $
How to prove the following result about the generalized hypergeometric function $_4 F_3$?
$$_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)\stackrel{?}=\frac{27}{16}\left(\pi^2-...
- 3,392
4
votes
1
answer
321
views
Prove that $\int_{0}^{1}x^2K^\prime(x)^3\text{d}x =\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2} -\frac{\pi^4}{40}$
I need to prove the following result
$$
\int_{0}^{1}x^2K^\prime(x)^3\text{d}x
=\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2}
-\frac{\pi^4}{40},
$$
where $K^\prime(x)=K\left(\sqrt{1-x^2}\right)$...
- 5,370
3
votes
1
answer
97
views
Is there any form of closed form solution for these two integrals?
I have two integrals of the from
\begin{equation}
\int_{-\pi/2}^{\pi/2} \mathrm d k e^{-i\omega \sin k} \frac{\lambda}{\alpha+\beta \cos 2k}
\end{equation}
and
\begin{equation}
\int_{-\pi/2}^{\pi/2} \...
- 712
7
votes
2
answers
411
views
Integrating $\int_{0}^{1} \frac{\arctan(x)\arctan(x^2)}{x^2} dx$
I found the following integral and wanted to know if there is a nice closed form solution in terms of elementary or some special functions (Polylogarithm, Clausen, etc).
$$\displaystyle \int_{0}^{1} \...
- 3,542
1
vote
1
answer
91
views
Closed form solution for $a^x - b^x = 1$
I'm looking for a closed form solution for $a^x - b^x = 1$ where $a,b\in \mathbb{R}$ are known and $x$ is unknown. There is a similar problem with positive sign instead of minus. Here are my attempts:
...
- 4,359
16
votes
1
answer
1k
views
On the integral $\int_0^1\frac{\arctan\sqrt{t^2+a}}{(t^2+b)\sqrt{t^2+a}}dt$
Let $0<b<a$ and define $$J(a,b)=\int_0^1\frac{\arctan\sqrt{t^2+a}}{(t^2+b)\sqrt{t^2+a}}dt.\tag1$$
I am seeking a closed form for $J(a,b)$.
I was motivated to find a closed form for $(1)$ after ...
- 17.3k
1
vote
2
answers
73
views
Closed-form solution to an infinite series?
I have the following series that I've confirmed on matlab to have some closed-form solution, but I can't find it through trial-and-error, and I definitely don't have the math background to just solve ...
- 21
1
vote
3
answers
200
views
What is the $n$ th derivative of $\ln(x)/(1+x^2)\:?$
I'm into something but I came across the problem of finding a closed form for
$$\left( \frac{\ln x}{1+x^2} \right)^{(n)}$$
where the little $(n)$ denotes the $n$ th derivative of the function. After ...
- 2,691
6
votes
2
answers
247
views
Closed-form expression for $F(x,y) = \int_0^1 \frac{\sqrt{t(1-t)}}{(t+x)^2 (t+y)} \mathrm{d}t$?
I am considering the following function
$$F(x,y) = \int_0^1 \frac{\sqrt{t(1-t)}}{(t+x)^2 (t+y)} \mathrm{d}t,$$
which is well-defined for any $x > 0$ and $y \geq 0$.
Is there a hope to obtain a ...
- 822
2
votes
0
answers
83
views
$ \int_R | \frac{ \cos(x) + \cos( \sqrt 2 x) + \cos( \sqrt 3 x)}{1 + x^2} | dx = ? $
Integrals where the integrand contains an absolute value can be very hard or impossible to express in closed form.
Computing the zero’s of the integrand might help.
But what if the zero’s have no ...
- 16.4k
4
votes
3
answers
243
views
What's the generalization for $\zeta(2n)$?
We know that
$$\zeta(2)=\frac{\pi^2}{6}\tag1$$
$$\zeta(4)=\frac{\pi^4}{90}
\tag2$$
$$\zeta(6)=\frac{\pi^6}{945}\tag3$$
$$.$$
$$.$$
$$.$$
My question is there a generalization for $\zeta(2n)$ in ...
- 25.8k