Questions tagged [constants]
For questions about mathematical constants, that are "significantly interesting in some way".
1
vote
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
votes
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 ...
0
votes
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 $N=1000$ ...
1
vote
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 ? ...
0
votes
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}\,)$$
I find $...
0
votes
2answers
41 views
How do you compute the singular series?
Terence Tao gives at his blog the following formula for something called the singular series:
$$\large\mathfrak{S}(h)=2\Pi_{2}\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$
where $\Pi_{2}=0.66016...$ is ...
0
votes
1answer
14 views
Redefinition of a Constant Leading to Nullification of Absolute Value
I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
0
votes
2answers
51 views
Computing the error function for Euler's number
By the error function for the sum $$\sum_{i = 0}^\infty \frac{1}{i!},$$ I mean the function $$f : \mathbb{R}_{> 0} \rightarrow \mathbb{N}$$ defined as follows.
For each $\varepsilon \in \mathbb{R}...
0
votes
1answer
28 views
Sum dissapearing when we assume some elements to be constant over time
I have the dividend discount model, which is the following expression:
$$
P_{j,t} = \sum_{\tau=1}^{\infty}D_\tau(1+g)^\tau(1+r)^{-\tau}=\frac{D_{\tau+1}}{r-g}
$$
Where $D_{\tau}$, is the dividend at ...
0
votes
1answer
56 views
Are constants from natural science of any importance at all in mathematics as real numbers?
In mathematics, people have discovered constants that are proven to be of great importance mathematically. For example,
Archimedes' constant, which is approximately $3.14159265$.
Euler's number, ...
1
vote
0answers
67 views
Find this inequality by finding the generalizating one too
Given $x,\,y,\,z$ such that $x,\,y,\,z\in [\,1,\,8\,]$$.$ 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 ...
0
votes
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 $\alpha$ ...
11
votes
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
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
0
votes
0answers
40 views
Convergence of series related to prime gaps and Gilbreath's conjecture
Gilbreath's conjecture states that when we take the absolute values of the consecutive differences of prime numbers, as so :
2, 3, 5, 7,11,13,17,19,23,...
1, 2, 2, 4, 2, 4, 2, 4,...
1, 0, 2, 2, 2, 2, ...
5
votes
4answers
205 views
Why does any function get thinner as $x$ is multiplied by a constant?
Example:
$$\cos(x)$$
$$\cos(8x)$$
"Thinner" might not be the correct term. But I just want to know why does changing $x$ to $8x$ make it look like that?
3
votes
1answer
86 views
Is the Pisot Triangle series known?
The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\...
6
votes
2answers
83 views
Why is Catalan's constant $G$ important?
I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related ...
6
votes
0answers
103 views
On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant
I. Intro
While trying to solve this post about the function,
$$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$
for $k=3$, I found out Mathematica can ...
4
votes
3answers
112 views
Have I discovered a new significance to a previously discovered constant?
I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged ...
0
votes
2answers
57 views
How many types of functions $f(x)$ satisfy $(f(x))^2$ + $(f'(x))^2 = constant$?
I know of two types of $f(x)$ that works.
First one is if $f(x) = c$, of course.
Second one is if it's a trigonometric function: $f(x) = c * \cos(x)$ or $f(x) = c * \sin(x)$.
My question is, are ...
5
votes
0answers
115 views
Is this (1.11716..) a known/named constant?
While looking at the Wikipedia entry for the Supergolden ratio aka Narayana's cow constant (cf. https://en.wikipedia.org/wiki/Supergolden_ratio), I noticed the following construction involving that ...
0
votes
1answer
29 views
Constant lengths (dimensional constants) and scaling
Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is ...
2
votes
2answers
62 views
Does the constant $C$ in this solution to a differential equation equal infinity?
The problem is $y' = -\frac{1}{t^2} - \frac{1}{t}y + y^2;\ y_p = \frac{1}{t}$. My solution is
$$\begin{align}
y = \frac{1}{t} + B &\implies y' = -\frac{1}{t^2} + B' \\
&\implies -\frac{1}{t^...
14
votes
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
$$G=\...
71
votes
1answer
4k views
What does the mysterious constant marked by C on a slide rule indicate?
Years ago, before everyone (or anyone) had electronic calculators, I had a pocket slide rule which I used in secondary school until the first TI-30 cane out.
Recently I dug it out. Here's a photo of ...
4
votes
2answers
161 views
Quadratic Formula With Independent and Dependent Variables
Given the differential equation $dy/dt = (y + t)^2$, we can apply the u-substitution $u = y + t$ to arrive at the separable differential equation $du/dt = u^2 + 1$. This separates to $1/(u^2 + 1)\ du ...
2
votes
1answer
79 views
Nature of constant $c$ from $\lim\limits_{n\to\infty}\left(\sum\limits_{k=1}^{n}(1/k)^{1/k}-n+\frac{\log^2(n)}{2}-1\right)=c$
If we take
$$\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$$
What is the nature of constant $c$? Is it really constant (maybe it ...
2
votes
1answer
27 views
Finding a constant $B > 0 $, so that $\left\lVert x \right\rVert$ $\leq$ $ B* \left\lVert x \right\rVert_\infty$
How do I find a constant $B > 0 $, so that $\left\lVert x \right\rVert$ $\leq$ $ B* \left\lVert x \right\rVert_\infty$ works for all $x \in \mathbb{R}^n$?
I am not sure but I think I have to look ...
-1
votes
3answers
52 views
Non-linear first order ODE with auxiliary variable
I have a problem with this equation: $y'(x)=\frac{2y(x)-x}{2x-y(x)}$.
Using $y=xz$ i'm arrived to prove that $\frac{z-1}{(z+1)^{3}}=e^{2c}x^{2}$, but now i'm stuck.
How can i explain the $z$? I've ...
0
votes
0answers
28 views
Very small constants from poly-bernoulli
If we define
$$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,...
1
vote
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?
0
votes
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
votes
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!
$$\lim_{x\rightarrow \infty}(\sqrt{x^2+...
0
votes
0answers
56 views
Brun’s constant and irrational numbers
It is trivial that if there are finitely many twin primes then Brun’s constant must be a rational number. And GammaTester (below) has offered an example of an infinite series that converges to a ...
0
votes
1answer
80 views
What value does the continued fraction $[1,2,3,1,2,3,…]$ represent?
I have found a number that satisfies this continued fraction:$$n=1+{1\over{2+{1\over{3+{1\over n}}}}}$$ With a value of about $1.4403$ after 9 layers of nesting. I've tried googling it and plugging it ...
2
votes
1answer
72 views
Explicit algorithms and algorithms involving unknowns
Let's assume the you have two algorithms for computing some single but complicated number (e.g., the Ramsey number $R(5,5)$). Both are provided as high-level, semi-formal textbook descriptions.
The ...
2
votes
1answer
161 views
Proving a sum identity for the Bernoulli numbers
I would like to prove that
$$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)$$
I've checked that it holds for n=2-20. This sum came up in a pet ...
5
votes
1answer
203 views
How are values of the Dirichlet Beta function derivative derived?
Wolfram Mathworld gives the following values for the beta function derivative.
$$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \...
2
votes
2answers
97 views
Pull constant out of summation
In the solution of my homework there's this step that I don't understand:
$$\begin{align}\mathsf {Var}[X] & = \sum_{k=1}^{n}\dfrac{n(k-1)}{(n-k+1)^2}\\ &= n \sum_{k=1}^{n}\dfrac{(n-k)}{k^2}\\&...
-4
votes
2answers
36 views
product rule constants
I'm struggling with the concept of constants in the context of derivatives.
For example;
$$f(\theta)=r(\cos\theta-1)$$ where $r$ is a constant.
$$f'(\theta)=-r\sin\theta$$
Why is the product rule ...
0
votes
3answers
91 views
Which one is bigger, $e ^ {3 \pi}$ or $ \space 3 ^ {e \pi}$? [duplicate]
I have this equation, and I've tried checking which is bigger,
$ e ^ {3 \pi} \space ? \space 3 ^ {e \pi}$
What I've tried:
$ e ^ {3 \pi} \space ? \space 3 ^ {\pi e} / \sqrt[\pi]{} $
$ e ^ {3} \...
0
votes
2answers
29 views
Given density function finding constant
Given X, a continuous random variable and the density:
$$
f(x) =
\begin{cases}
x^2, & \text{if $x$ $\in$ [-1,1] ,} \\
c \cdot \frac{1}{|x|^k}, & \text{else}
\end{cases}
$$
Where $c\in \...
4
votes
0answers
159 views
Is there a name for this “Collatz constant”?
Right now I'm calling a convergent number based on the Collatz conjecture the "Collatz constant". I'm wondering if it have an ACTUAL name? And if it actually converges?
Details
The Collatz ...
0
votes
3answers
174 views
Why does the constant of integration move?
I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from:
$$y = \exp(-\cos x + C)$$
To:
$$y = A\cdot \exp(-\cos x)$$
I understand the constant of ...
1
vote
2answers
74 views
Recursive formula for e
Looking for a finite recursive formula for the constant e preferably using standard operators (ones a calculator could carry out)
i.e. a formula of the form $ x_{n+1}=f(x_n)$ where $\lim_{n\...
0
votes
1answer
56 views
Behavior of $\int_{-\infty}^{a} \tan (e^x)dx$ for $a < 0$
I have tried to give a closed form of $\int_{-\infty}^{a} \tan (e^x)dx$ using $\text{Si}(x)$ ( sine integral of x) and $\text{Ci}(x)$ ( cosine integral of $x$) , but I didn't succed really the ...
3
votes
2answers
500 views
Constant random variable
How do I plot the cumulative distribution function and probability mass function of the constant random variable $X(\omega)=2$ for all $\omega$?
1
vote
1answer
102 views
On questions about the integral of the logarithm of the Riemann zeta function II
With the help of Wolfram Alpha online calculator I know good approximations of the real part of the integral $$\int_0^1 \log \left(\zeta(x)\right)dx,$$
where $\zeta(x)$ denotes the Riemann zeta ...
-3
votes
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
votes
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. ...
