Questions tagged [closed-form]
A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
3,683
questions
4
votes
0
answers
83
views
Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.4k
0
votes
0
answers
52
views
how to use Gauss Multiplication Formula for Gamma function?
I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$
$$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$
but I didn't ...
- 1,577
-1
votes
0
answers
58
views
Closed form of summation involving square root [closed]
Does anyone have any idea how to find a closed form for the following summation? I cannot for the life of me figure anything out.
$$\sum_{i=1}^n(n-i)\sqrt{i^2+1}$$
0
votes
0
answers
45
views
Closed form for nested sum involving ratios of binomial coefficients
I ran into the following nested sum of binomial coefficients in my research, but I couldn't find the closed form expression for it. I looked at various sources and still couldn't find the answer. So I ...
0
votes
1
answer
66
views
Solution to "basic" ODE which came up during research
So during my research this ODE came up, but I'm not very experienced in explicitly solving ODEs. Does this one have a known explicit solution
$$
t\,y^{\prime}(t) = (bt+k)y(t)
$$
where $b,k$ are non-...
- 6,779
2
votes
1
answer
91
views
Closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$?
Is there a closed form for
$A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$
??
We know
$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {...
- 16.3k
0
votes
0
answers
33
views
Closed form of $f(n) = \prod_{m=2}^{n-1}( e^{\pi i n/m} - e^{-\pi i n/m})$
For context, I am not a mathematician, but I like to do explore some math concepts a couple of times a year. I have been playing with an idea for the past few years or so and a while back I asked this ...
- 131
1
vote
0
answers
72
views
Is there a closed form for the following integral?
I want to find a closed form of the following integral:
$$
I \equiv \int_{0}^{R}\frac{b\operatorname{J}_{1}\left(ax\right) \operatorname{J}_{0}\left(bx\right) + a\operatorname{J}_{1}\left(bx\right)\...
- 109
2
votes
1
answer
66
views
Closed form for $\sum \left (\pm a_1 \pm a_2 \pm \dots \pm a_n\right )^\ell$
I realized that if you take the $2^n$ quantities
$$\pm a_1 \pm a_2 \pm \dots \pm a_n$$
and consider the sum of their squares, then the product terms cancel out nicely to give
$$\sum \left (\pm a_1 \pm ...
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
9
votes
3
answers
2k
views
How does Wolfram Alpha know this closed form?
I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function:
$$
\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).
$$
How does WA know ...
- 5,285
4
votes
2
answers
214
views
Evaluating $ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $
The remarkable Ramanujan nested radical is
$ \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3 $
What can be said about
$ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $ ?
With Mathematica I found that ...
- 843
3
votes
1
answer
134
views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.
We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 843
0
votes
1
answer
43
views
Why $\theta ([\xi_f, \xi_g])=\frac{1}{2}\xi_f \langle \theta ,\xi_g\rangle -\frac{1}{2}\xi_g \langle \theta ,\xi_f\rangle$?
Let $(M,\omega)$ be a symplextic manifold with $\omega =-d\theta$. For $f,g\in C^{\infty}(M)$, we can define a Poisson bracket $\{ f,g\}=\omega (\xi_{f},\xi_f)$ where $i_{\xi_f}\omega=\omega(\xi_f,\...
- 755
1
vote
0
answers
45
views
In search for references of exact or closed-form solutions of the spinning heavy symmetric top
The system is depicted by the following figure.
Characteristic fixed parameters are:
$m$ for mass;
$g$ for gravity constant;
$\ell$ is the distance between the contact point and the center of mass;
...
- 435
5
votes
1
answer
157
views
Closed form of $ \int_0^\infty dx r J_1(r x) \left [ J_0(x) \right ]^Q $
I'm looking for a closed form for the following definite integral
$$
I(r,Q) := r \int_0^\infty dx J_1(r x) \left [ J_0(x) \right ]^Q
$$
where $r$ is a positive real, $Q$ is a positive integer and $J_a$...
- 2,624
0
votes
0
answers
39
views
How to show that $\sum_{j = 1}^{A \left({n}\right)} \left\{{\frac{n - {j}^{2}}{2\, j}}\right\}$ satisfies Weyl's criterion.
We can write this sum in terms of even ($j = 2k$) and odd ($j = 2k-1$) summation index as
$$\sum_{j = 1}^{A \left({n}\right)} \left\{{\frac{n - {j}^{2}}{2\, j}}\right\} = \sum_{k = 1}^{\lfloor{A \left(...
- 957
3
votes
2
answers
112
views
Is it possible to find the $n$th derivative of Gamma function?
By repeatedly differentiating $\Gamma(x)$, I noticed that
$$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$
where $\psi^{(a)}(x)$ is ...
- 25.8k
1
vote
3
answers
114
views
Is there any nonlinear function $f(x)$ for which both $\sin (f(x))$ and $\cos(f(x))$ have closed-form integrals?
I'm interested in closed-form antiderivaties of elementary functions.
The question arises from seeing that something as simple as $\int\sin(x^2)dx$ and $\int\cos(x^2)dx$ generates the Fresnel ...
1
vote
2
answers
145
views
Is there a closed form for the integral $\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$, where $k$ is a natural number?
To start, I am aware that our integral $I(k)=\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$ is equal to $$I(k)=\int_{0}^{\pi/4}x^{k}\ln\sin x\, dx-\int_{0}^{\pi/4}x^{k}\ln\cos x\, dx$$, but I cannot seem to ...
- 586
0
votes
2
answers
139
views
How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{4y \, da \, dy}{(y^2 + ay + 1)(y^2 - ay + 1)}$? [duplicate]
Question: How to evaluate $$\int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \sin x} \right) \, dx$$
My attempt
The original integral is:
$$
J = \int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \...
1
vote
2
answers
81
views
Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 31
8
votes
1
answer
234
views
How to solve $\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}dx$
One of my friends sent me a list of integrals (all without solutions ) one of those problems is: $$\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}...
- 6,352
8
votes
3
answers
260
views
How to evaluate $\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx$?
Question: How to evaluate $$\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx?$$
My attempt
We will denote the main integral as $\Omega$.
$$\Omega=\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, ...
0
votes
0
answers
32
views
What is the power series for $K_{\lambda}(w) = \int_{0}^{\infty}z^{\lambda-1}\exp\left\{-\frac{w}{2}\left(z + \frac{1}{z}\right)\right\}\mathrm{d}z$?
As the title says, I would like to know what is the power series for (or at least a good approximation)
\begin{align*}
K_{\lambda}(w) = \frac{1}{2}\int_{0}^{\infty}z^{\lambda-1}\exp\left\{-\frac{w}{2}\...
- 31
8
votes
1
answer
174
views
how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?
I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps.
or is it possible to evaluate $$\int_0^1 \ln^n(...
4
votes
3
answers
145
views
How to Find closed form $\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$
How to Find closed form :$$\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$$ where $a_n=\sum_{k=1}^{2n-1}\frac{(-1)^{k-1}}{k}$
$$S(x)=\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n=\sum_{n=1}^{\infty}\int^1_0y^{n-1}x^{...
- 2,288
5
votes
2
answers
190
views
The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?
I am looking for a closed form for
$$
R = \frac{1}{2}\,\exp\left(-\int_{0}^{1}
\log\left(\sin\left(\frac{\pi}{6} +
\frac{2\pi}{3}\,x\right)\right){\rm d}x\right)\approx 0.6159
$$
Wolfram does not give ...
- 25.6k
5
votes
2
answers
237
views
How can I show that $\int_0^{\frac{\pi}{2}}\sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx$.
Question: How can I show that
\begin{align}
& \int_0^{\frac{\pi}{2}} \sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right)\,dx
\\[2mm] = & \
{\small\log\left(\left(2\sqrt{2-\sqrt{2}}...
0
votes
1
answer
43
views
Is there a closed form method of expressing the *content* of integer partitions of $n$?
I know that the question of a closed form for the number of partitions of $n$, often written $p(n)$, is an open one (perhaps answered by the paper referred to in this question's answer, although I'm ...
