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".
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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:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
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how to use Gauss Multiplication Formula for Gamma function?
I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$
$$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$
but I didn't ...
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Closed form of summation involving square root [closed]
Does anyone have any idea how to find a closed form for the following summation? I cannot for the life of me figure anything out.
$$\sum_{i=1}^n(n-i)\sqrt{i^2+1}$$
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Closed form solution for $\displaystyle \sum _{i=r+1}^{k}\frac{1}{i-1} \cdot \frac{( n-i) !}{( k-i) !}$, where $n,k,r$ are constants and $r \leq k<n$ [closed]
Context
I arrived at this summation, while computing a probability of The Secretary Problem.
Question:
Find Closed form for this or if possible simplify
$$
\frac{r}{n}\sum_{i = r + 1}^{k}\frac{1}{i - ...
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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 ...
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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-...
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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
$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {...
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Closed form of $f(n) = \prod_{m=2}^{n-1}( e^{\pi i n/m} - e^{-\pi i n/m})$
For context, I am not a mathematician, but I like to do explore some math concepts a couple of times a year. I have been playing with an idea for the past few years or so and a while back I asked this ...
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Is there a closed form for the following integral?
I want to find a closed form of the following integral:
$$
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)\...
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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
$$\sum \left (\pm a_1 \pm ...
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Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
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How does Wolfram Alpha know this closed form?
I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function:
$$
\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).
$$
How does WA know ...
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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 ...
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answer
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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 ...
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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,\...
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