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Questions tagged [constants]

For questions about mathematical constants, that are "significantly interesting in some way".

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1answer
38 views

Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
0
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1answer
24 views

Inverse Laplace Transform - Pulling out the constant

If you refer to my picture: https://i.stack.imgur.com/lVsU1.png I'm having a hard time understanding why in the 2nd step the fraction is split up in two terms when 2 is a constant. I get why you ...
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1answer
41 views

What constant $c$ will make $ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N ?$

What constant $c$ will make this equality valid for any $N$ chosen? $$ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N. $$ I tried getting a rough idea of what $c$ should be and got about $1.46$ when $N=1000$ ...
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0answers
25 views

Question about the Gamma function

My question is fairly simple: I was wondering if $\,\,\,\Gamma\left(\pi\right) = 2.2880377\ldots\,\,\,$ had any special meaning. Is it irrational ?. transcendental ? is it useless ? ...
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1answer
68 views

An inequality with the $constant= \frac{1}{2}+ \frac{5}{18}\,\sqrt{3}$

Given $a,\,b,\,c> 0$ such that$:$ $a+ b+ c= 3$$.$ Prove$:$ $$\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}\geqq \left ( \frac{1}{2}+ \frac{5}{18}\,\sqrt{3} \right )(\,a^{\,2}+ b^{\,2}+ c^{\,2}\,)$$ I find $...
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2answers
41 views

How do you compute the singular series?

Terence Tao gives at his blog the following formula for something called the singular series: $$\large\mathfrak{S}(h)=2\Pi_{2}\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$ where $\Pi_{2}=0.66016...$ is ...
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1answer
14 views

Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
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2answers
51 views

Computing the error function for Euler's number

By the error function for the sum $$\sum_{i = 0}^\infty \frac{1}{i!},$$ I mean the function $$f : \mathbb{R}_{> 0} \rightarrow \mathbb{N}$$ defined as follows. For each $\varepsilon \in \mathbb{R}...
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1answer
28 views

Sum dissapearing when we assume some elements to be constant over time

I have the dividend discount model, which is the following expression: $$ P_{j,t} = \sum_{\tau=1}^{\infty}D_\tau(1+g)^\tau(1+r)^{-\tau}=\frac{D_{\tau+1}}{r-g} $$ Where $D_{\tau}$, is the dividend at ...
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1answer
56 views

Are constants from natural science of any importance at all in mathematics as real numbers?

In mathematics, people have discovered constants that are proven to be of great importance mathematically. For example, Archimedes' constant, which is approximately $3.14159265$. Euler's number, ...
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0answers
67 views

Find this inequality by finding the generalizating one too

Given $x,\,y,\,z$ such that $x,\,y,\,z\in [\,1,\,8\,]$$.$ Prove$:$ $$\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$$ By computer$,$ I have found ...
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1answer
38 views

Is it possible to pull constant out of this summation

I'm trying to prove the following: $$ F(\alpha x) = \sum_{m=1}^M \bigl( \sum_{i=1}^N (\alpha x_i)^{\omega_m}\bigl)^{1/{\omega_m}} = \alpha F(x) $$ But I can't quite understand how to pull $\alpha$ ...
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1answer
253 views

product= $\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\left(\frac{11^{11}3^3}{13^{13}}\right)^{1/20}\sqrt{\frac{3}{7^{7/6}\pi}\sqrt{\frac2\pi}}$

$\mathrm G$ is Catalan's constant. I recently found the product $$ \alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
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0answers
40 views

Convergence of series related to prime gaps and Gilbreath's conjecture

Gilbreath's conjecture states that when we take the absolute values of the consecutive differences of prime numbers, as so : 2, 3, 5, 7,11,13,17,19,23,... 1, 2, 2, 4, 2, 4, 2, 4,... 1, 0, 2, 2, 2, 2, ...
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4answers
205 views

Why does any function get thinner as $x$ is multiplied by a constant?

Example: $$\cos(x)$$ $$\cos(8x)$$ "Thinner" might not be the correct term. But I just want to know why does changing $x$ to $8x$ make it look like that?
3
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1answer
86 views

Is the Pisot Triangle series known?

The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\...
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2answers
83 views

Why is Catalan's constant $G$ important?

I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related ...
6
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0answers
103 views

On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant

I. Intro While trying to solve this post about the function, $$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$ for $k=3$, I found out Mathematica can ...
4
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3answers
112 views

Have I discovered a new significance to a previously discovered constant?

I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged ...
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2answers
57 views

How many types of functions $f(x)$ satisfy $(f(x))^2$ + $(f'(x))^2 = constant$?

I know of two types of $f(x)$ that works. First one is if $f(x) = c$, of course. Second one is if it's a trigonometric function: $f(x) = c * \cos(x)$ or $f(x) = c * \sin(x)$. My question is, are ...
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0answers
115 views

Is this (1.11716..) a known/named constant?

While looking at the Wikipedia entry for the Supergolden ratio aka Narayana's cow constant (cf. https://en.wikipedia.org/wiki/Supergolden_ratio), I noticed the following construction involving that ...
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1answer
29 views

Constant lengths (dimensional constants) and scaling

Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is ...
2
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2answers
62 views

Does the constant $C$ in this solution to a differential equation equal infinity?

The problem is $y' = -\frac{1}{t^2} - \frac{1}{t}y + y^2;\ y_p = \frac{1}{t}$. My solution is $$\begin{align} y = \frac{1}{t} + B &\implies y' = -\frac{1}{t^2} + B' \\ &\implies -\frac{1}{t^...
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5answers
283 views

Relationship between Catalan's constant and $\pi$

How related are $G$ (Catalan's constant) and $\pi$? I seem to encounter $G$ a lot when computing definite integrals involving logarithms and trig functions. Example: It is well known that $$G=\...
71
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1answer
4k views

What does the mysterious constant marked by C on a slide rule indicate?

Years ago, before everyone (or anyone) had electronic calculators, I had a pocket slide rule which I used in secondary school until the first TI-30 cane out. Recently I dug it out. Here's a photo of ...
4
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2answers
161 views

Quadratic Formula With Independent and Dependent Variables

Given the differential equation $dy/dt = (y + t)^2$, we can apply the u-substitution $u = y + t$ to arrive at the separable differential equation $du/dt = u^2 + 1$. This separates to $1/(u^2 + 1)\ du ...
2
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1answer
79 views

Nature of constant $c$ from $\lim\limits_{n\to\infty}\left(\sum\limits_{k=1}^{n}(1/k)^{1/k}-n+\frac{\log^2(n)}{2}-1\right)=c$

If we take $$\sum\limits_{k=1}^{n}(1/k)^{1/k}=a(n)$$ so $$\lim\limits_{n\to\infty}\left(a(n)-n+\frac{\log^2(n)}{2}-1\right)=c$$ What is the nature of constant $c$? Is it really constant (maybe it ...
2
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1answer
27 views

Finding a constant $B > 0 $, so that $\left\lVert x \right\rVert$ $\leq$ $ B* \left\lVert x \right\rVert_\infty$

How do I find a constant $B > 0 $, so that $\left\lVert x \right\rVert$ $\leq$ $ B* \left\lVert x \right\rVert_\infty$ works for all $x \in \mathbb{R}^n$? I am not sure but I think I have to look ...
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3answers
52 views

Non-linear first order ODE with auxiliary variable

I have a problem with this equation: $y'(x)=\frac{2y(x)-x}{2x-y(x)}$. Using $y=xz$ i'm arrived to prove that $\frac{z-1}{(z+1)^{3}}=e^{2c}x^{2}$, but now i'm stuck. How can i explain the $z$? I've ...
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0answers
28 views

Very small constants from poly-bernoulli

If we define $$a_{n}(m)=\sum\limits_{k=0}^{n}k!{n\brace k}(k+1)^m(-1)^{n-k}$$ $$\prod\limits_{k=2}^{n+1}1-kx=\sum\limits_{k=0}^{n}t(n,k)x^k$$ for $n>0$, $m\geqslant0$, so $$\sum\limits_{k=0}^{n}t(n,...
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1answer
83 views

Are all constant fields conservative? [closed]

Are all constant fields conservative? Can there be some constant vector fields which are not conservative?
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3answers
61 views

Are there any known plastic constant triangles?

I am trying to determine if there are any known plastic constant triangles. By this I mean specifically triangles for which all sides are powers of the plastic constant, $p\approx1.324717957244746$. ...
2
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2answers
49 views

For what values of $a$ and $b$ does $\lim_{x\rightarrow \infty}(\sqrt{x^2+x+1}-ax-b)=1$?

I have a question about limits tending to infinity. I need to find the constants $a$ and $b$ for which this limit takes the value 1. Please, help! Thank you! $$\lim_{x\rightarrow \infty}(\sqrt{x^2+...
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0answers
56 views

Brun’s constant and irrational numbers

It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a ...
0
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1answer
80 views

What value does the continued fraction $[1,2,3,1,2,3,…]$ represent?

I have found a number that satisfies this continued fraction:$$n=1+{1\over{2+{1\over{3+{1\over n}}}}}$$ With a value of about $1.4403$ after 9 layers of nesting. I've tried googling it and plugging it ...
2
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1answer
72 views

Explicit algorithms and algorithms involving unknowns

Let's assume the you have two algorithms for computing some single but complicated number (e.g., the Ramsey number $R(5,5)$). Both are provided as high-level, semi-formal textbook descriptions. The ...
2
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1answer
161 views

Proving a sum identity for the Bernoulli numbers

I would like to prove that $$B_{2n}=\frac{1}{2(2n-1)(2^{2n}-1)}\left(1-\sum_{k=1}^{n-1}(2k-1)2^{2k}\binom{2n}{2k}B_{2k}\right)$$ I've checked that it holds for n=2-20. This sum came up in a pet ...
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1answer
203 views

How are values of the Dirichlet Beta function derivative derived?

Wolfram Mathworld gives the following values for the beta function derivative. $$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \...
2
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2answers
97 views

Pull constant out of summation

In the solution of my homework there's this step that I don't understand: $$\begin{align}\mathsf {Var}[X] & = \sum_{k=1}^{n}\dfrac{n(k-1)}{(n-k+1)^2}\\ &= n \sum_{k=1}^{n}\dfrac{(n-k)}{k^2}\\&...
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2answers
36 views

product rule constants

I'm struggling with the concept of constants in the context of derivatives. For example; $$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant. $$f'(\theta)=-r\sin\theta$$ Why is the product rule ...
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3answers
91 views

Which one is bigger, $e ^ {3 \pi}$ or $ \space 3 ^ {e \pi}$? [duplicate]

I have this equation, and I've tried checking which is bigger, $ e ^ {3 \pi} \space ? \space 3 ^ {e \pi}$ What I've tried: $ e ^ {3 \pi} \space ? \space 3 ^ {\pi e} / \sqrt[\pi]{} $ $ e ^ {3} \...
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2answers
29 views

Given density function finding constant

Given X, a continuous random variable and the density: $$ f(x) = \begin{cases} x^2, & \text{if $x$ $\in$ [-1,1] ,} \\ c \cdot \frac{1}{|x|^k}, & \text{else} \end{cases} $$ Where $c\in \...
4
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0answers
159 views

Is there a name for this “Collatz constant”?

Right now I'm calling a convergent number based on the Collatz conjecture the "Collatz constant". I'm wondering if it have an ACTUAL name? And if it actually converges? Details The Collatz ...
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3answers
174 views

Why does the constant of integration move?

I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from: $$y = \exp(-\cos x + C)$$ To: $$y = A\cdot \exp(-\cos x)$$ I understand the constant of ...
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2answers
74 views

Recursive formula for e

Looking for a finite recursive formula for the constant e preferably using standard operators (ones a calculator could carry out) i.e. a formula of the form $ x_{n+1}=f(x_n)$ where $\lim_{n\...
0
votes
1answer
56 views

Behavior of $\int_{-\infty}^{a} \tan (e^x)dx$ for $a < 0$

I have tried to give a closed form of $\int_{-\infty}^{a} \tan (e^x)dx$ using $\text{Si}(x)$ ( sine integral of x) and $\text{Ci}(x)$ ( cosine integral of $x$) , but I didn't succed really the ...
3
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2answers
500 views

Constant random variable

How do I plot the cumulative distribution function and probability mass function of the constant random variable $X(\omega)=2$ for all $\omega$?
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1answer
102 views

On questions about the integral of the logarithm of the Riemann zeta function II

With the help of Wolfram Alpha online calculator I know good approximations of the real part of the integral $$\int_0^1 \log \left(\zeta(x)\right)dx,$$ where $\zeta(x)$ denotes the Riemann zeta ...
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3answers
239 views

Is infinity constant? [closed]

Infinity while being an infinite set or limit is always defined as +∞ or -∞ which can literally be any subset of numbers: ...
0
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0answers
23 views

a question about solving recurrence with constant

Solve ${x_n}^2sin(\theta)={d^2-\frac{|x_{n-1}-x_{n+1}|}4}^2{} \& x_{n+1}=x_ncos(\theta)\pm \sqrt{{x_n}^2(cos^2(\theta)+1)-d^2}$ --I am really confused about howcan I deal with the constant term. ...