Questions tagged [constants]
For questions about mathematical constants, that are "significantly interesting in some way".
589 questions
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An efficient series for the Glaisher–Kinkelin constant?
From the following identity:
$$ 3\ln A-\frac{1}{4}-\frac{1}{3}\ln 2=\int_0^1\frac{1}{\ln z}\left(\frac{1}{4}-\frac{1}{(1+z)^2}\right)dz$$
Is it viable to obtain a series expansion for $\ln(A)$? $A$ ...
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Optimal constant in Sobolev embedding $H^1(0,s) \subset L^\infty$
The paper by Marti Evaluation of the Least Constant in Sobolev's Inequality (Marti) for $H^1(0,s)$ has a sketch of a simple fundamental theorem of calculus computation to obtain a less optimal ...
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1
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Simple question about $\int_0^{1} (1-x^n)^{\frac{1}{n}} dx$
I'm not sure if it's correct to ask this question, since it may be too obvious, but here we are. I am very superficially familiar with integrals and decided to just have fun by substituting random ...
user929945
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2
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Irrationality of Euler's constant: where is my proof wrong? [closed]
I've had another look into Euler's constant since my last post and developed the argument below. I'm new to number theory, so am wondering where this might go astray? Can someone point out the error ...
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Is Euler's constant expressible in terms of Pi? [closed]
My question boils down to whether I can rewrite the sum of the Harmonic series in particular way?
Consider Euler's constant, defined as the difference between the Harmonic series and natural logarithm....
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4
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2
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Like the supergolden triangle, can we construct a supersilver triangle?
I. Golden triangle
The golden triangle has the ratio $\phi=\dfrac{a}{b}=\dfrac{1+\sqrt5}2$. We can assume all sides as powers of $\phi$, so $a=\phi$ and $b=\phi^0=1$. The apex angle is then $\theta=...
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Does the geometric mean of the coefficients always converge in this generalization of continued fractions?
Every number has a canonical form as a simple continued fraction; that is, for all $x$ there exist unique $a_n$ such that
$$x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}.$$
It is ...
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What convergence test to use for $\sum( 1/ P_n)$ where $P_n$ is the maximal prime gap primes?
I know that $1/p_k$ diverges slowly. https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes
This sequence is different. By choosing only to sum the ends of the maximal ...
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The cubic relationship of three consecutive terms of tribonacci, Padovan, and Narayana sequences?
I. Fibonacci and Lucas numbers
$$F_n = 1,1,2,3,5,8,13,\dots\\[7pt]
L_n = 2, 1, 3, 4, 7, 11, 18\dots$$
There is a relationship between two consecutive terms such that,
$$M(x,y) = x^2+xy-y^2$$
$$M(F_n,\,...
- 61.3k
6
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1
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Strange symbol from Wolfram Alpha when finding closed-form solution of a number
I am trying to find an exact symbolic representation of a numerical solution to a transcendental equation:
$$ \cosh{x} = e^{0.5 x} $$
I know the numerical solution is approximately
$$1.218755726872$$
...
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About constant of Decoupling inequality
As an article says,
let $F_I:=\mathcal{F}^{-1}\left(\widehat{F} \cdot \mathbb{1}_{\Omega_I(\delta)}\right)$, then $$\left\|\sum_{I \subset[-1,1], \ell(I)=\delta} F_I\right\|_{L^p\left(\mathbb{R}^2\...
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Prove Integral Evaluates to Fransén–Robinson constant
How can I prove the following identity,
$$\int_{1}^{\infty} \frac{1}{\pi^2x^2 + x^2\log^2\log(x)} \mathrm dx = F-e$$
where $F$ is the Fransén–Robinson constant.
I have verified the result numerically ...
user1295974
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Proving that the arithmetic-geometric mean of $1$ and $\sqrt{2}$ is $\pi/\varpi$, where $\varpi$ is the lemniscate constant
I have just read on Wikipedia that in 1799 Gauss proved that $$\color {blue}{AGM(1,\sqrt2)=\frac{\pi}{\varpi},}$$ where $AGM$ is the arithmetic–geometric mean (see the previous link) and $\varpi$ is ...
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Why this determinant is equal to 1/2025?
I have seen this picture (I don't know the source):
And verified by Mathematica, there are a little difference ($6.82535\times 10^{-7}$), and I'm not sure if it is machine error.
...
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Let $\gamma=\lim_{k\rightarrow1}\sum_{t=1}^\infty k\frac{(kt+k-1)(t+1)^{1/k}-k(t+1)t^{1/k}}{(k-1)t(t+1)}$, is it easier to see if $\gamma\in\Bbb{Q}$?
Does This Simplified Expression of the Euler-Mascheroni Constant Aid in Proving Its Rationality or Transcendence?
When I say "simplified", I mean that it doesn't contain Euler's number, the ...