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Questions tagged [constants]

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

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Matrix Norm Inequality: $c \sum_{k=1}^{n} \left| \sum_{l=1}^{n} \epsilon_{l} A_{k,l} \right| \geq \sum_{l=1}^{n} \sqrt{\sum_{k=1}^{n} A_{k,l}^{2}}$

Question Show that there is a real universal constant $c$ with the property that for all positive integers $n$, and all nonzero $n \times n$ real matrices $A$, there are signs $\epsilon_1, ..., \...
quhdshb's user avatar
  • 31
1 vote
0 answers
39 views

(Im)possibility of closed-form expression of Clausen functions

When I started learning Riemann zeta function, I was fascinated that $\zeta(2n)$ can be expressed with finite integers and $\pi$ while $\pi$ has no obvious relation with the sum-$\zeta(2n)$ but no &...
Quý Nhân Đặng Hoàng's user avatar
0 votes
0 answers
30 views

Algebra problems involving constant and coefficient of exponential equations

The function $f$ is defined by $f(x)=a(3.7^x+3.7^b),$ where $a$ and $b$ are integer constants and $0<a<b$. The functions $g$ and $h$ are equivalent to function $f$ where $k$ and $m$ are ...
user1328522's user avatar
5 votes
2 answers
193 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 ...
Dan's user avatar
  • 25.7k
0 votes
0 answers
39 views

Is $c = \frac{\sqrt 3}{4} \frac{\pi}{4} \prod_{p = 2 \mod 3} \sqrt{\frac{p^2}{p^2-1}} \prod_{q = u^2 + 3 v^2} \sqrt{\frac{q^2}{q^2-1}}$?

Consider the sum of $2$ squares and Gauss circle problem https://en.wikipedia.org/wiki/Gauss_circle_problem and also The Landau-Ramanujan Constant that relates to the sum of 2 squares. See : http://en....
mick's user avatar
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0 votes
0 answers
23 views

a cool optimization problem involving cubes surfaces and volumes

Consider a codimension one surface of revolution $S$ and an embedding $e:S \hookrightarrow X^n$ for $X^n=[0,1]^n$ with points $p,q$ elements of $\partial X^n$ where $\partial X^n=X^n-(0,1)^n$ for $\...
zeta space's user avatar
2 votes
1 answer
65 views

Is it "ok" to see constants as a "section" in $n$ dimentional space to get to lower dimentionality space?

Given that in $y=5x$, $5$ is a constant (let's call it $a=5$) in a 2d graph. I thought that it could be seen as a simplification of a more general case of some 3d space where $y$ is a function of both ...
Igor's user avatar
  • 91
2 votes
2 answers
54 views

Prove $\int_0^\infty (1-\exp(-\operatorname{Ei}(t)))dt=\int_0^\infty\exp(-t-\operatorname{Ei}(t))dt$

How to prove $$\int_0^\infty (1-\exp(-\operatorname{Ei}(t)))dt=\int_0^\infty\exp(-t-\operatorname{Ei}(t))dt$$ where $\operatorname{Ei}(t)=\int_1^\infty \frac{\exp(-xt)}{x} dx$? This constant value is ...
ueir's user avatar
  • 1,213
0 votes
0 answers
69 views

Function for the Champernowne constant that returns the digit position of any number.

In this paper, an equation is provided for locating the first occurance of any 10^n number: My question is if there is a generalized version of this equation that locates any number (eg, 314159), not ...
EternalPropagation's user avatar
5 votes
2 answers
451 views

Integration by parts does not work for this complex integral. Why?

There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted as it came across in research: $$ \frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-...
Тyma Gaidash's user avatar
25 votes
7 answers
1k views

Show by hand : $e^{e^2}>1000\phi$

Problem: Show by hand without any computer assistance: $$e^{e^2}>1000\phi,$$ where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$. I come across this limit showing: $$\...
Ranger-of-trente-deux-glands's user avatar
1 vote
2 answers
101 views

Are all constants of integration $C$ equal?

This is a question from very (very) basic calculus, but it concerns indefinite integrals and the constant $ C $ we always add when we find the antiderivative. This concerns some problems with ...
user avatar
1 vote
2 answers
149 views

Is a function $f$ defined on a closed interval $[a,b]$ constant, if $f'(x)=0$ for all $x∈(a,b)$ but $f'(a)$ or $f'(b)$ are nonzero real numbers?

Suppose we have a function $f$ that is defined on a closed interval $[a,b]$. The following can be proven from the Mean Value Theorem: If $f$ is continuous on the interval $[a,b]$ $f'(x)=0$ for all $...
CaptAngryEyes's user avatar
0 votes
0 answers
75 views

Differentiation over the hole PDE gives contradiction?

I have two PDE: $$\bigg(\frac{1}{R_1}\partial_rR_2+K(r)\bigg)+\bigg(-\frac{1}{S_1}\partial_\theta S_2-T(\theta)\bigg)+A(r)\sin\theta\frac{S_2}{S_1}=0\,, \\ \bigg(-\frac{1}{R_2}\partial_rR_1-K(r)\bigg)+...
Acephalus's user avatar
0 votes
0 answers
59 views

Bernstein uniform approximation by polynomials : What is $E_n(f)$?

I am having a difficult time understanding what Bernstein's constant is. Wikipedia states "Let $E_n(f)$ be the error of the best uniform approximation to a real function $f(x)$ in the interval $[-...
Kamal Saleh's user avatar
  • 6,549
-1 votes
1 answer
84 views

Dottie number and prime factorization

It's related to Dottie number and prime factorization . Let : $D=\operatorname{Dottie-number}\simeq 0.7390851332$ Now define $n\geq 3$ an integer: $$\lfloor\left(D\right)^{-2n}\rfloor=P$$ Then it ...
Ranger-of-trente-deux-glands's user avatar
1 vote
2 answers
97 views

Why the fixed elements under translation transformation $f(x) \mapsto f(x+1)$ of a field of rational functions must be constants?

Let $F$ be a field, and let $E=F(x)$ be the field of rational functions over $F$. We can see that the mapping $f(x) \mapsto f(x+1)$ is an automorphism of the field $E$. My question is, how to prove ...
David Lee's user avatar
1 vote
0 answers
85 views

Introducing a new constant $\alpha$ this way, what will be its properties? [closed]

Introduction First of all, let us introduce a concept of numerocity of a set, starting from the subsets of integers. $N(S)=\sum_{k=-\infty}^\infty p_s(k)=p_s(0)+\sum_{k=1}^\infty p_s(k)+\sum_{k=1}^\...
Anixx's user avatar
  • 1
2 votes
1 answer
89 views

Examples of famous constants that turned out to be rational

I was just reading through some random articles on the proofs (or lackthere of) of some famous constants being irrational or transcendental. (Such as $ \pi, e$, the Euler-Mascheroni constant etc.) ...
Carter Giese's user avatar
2 votes
1 answer
132 views

Is there a name for this constant and its value where $\alpha = \sum_{p}\frac{\log \left({p}\right)}{p \left({p-1}\right)}$

The constant $$\alpha = \sum_{p} \frac{\log \left({p}\right)}{p \left({p-1}\right)}$$ comes from the calculation $$\sum_{p=2}^{x} \frac{\log \left({p}\right)}{p-1} = \sum_{p=2}^{x} \frac{\log \left({p}...
Lorenz H Menke's user avatar
0 votes
0 answers
91 views

Proof of $\lim_{n\to \infty}(1 + \frac{b}{n})^n = e^b$ [duplicate]

I found one definition of $e$ to be $\lim_{n\to \infty}(1 + \frac{1}{n})^n = e$ and I checked it with a graphing calculator to be true. However the $1$ in the numerator is a special case, where it ...
Gustamons's user avatar
0 votes
0 answers
97 views

Searching for a paper on "$e$ is not expressible as a simple integral"

I have seen a paper whose main aim was to explain a conjecture that $e$ is not expressible as the integral of a "simple function" over a "simple" region. In contrast to $\pi$ ...
Maesumi's user avatar
  • 3,702
-1 votes
1 answer
53 views

Constant Higher-order derivatives

Given the equation $$\frac{d^ky}{dx^k}=c$$ where $k$ represents the number of higher-order derivatives and $c$ represents any constant real number except $0$. Eg : $$\frac{d^{11}y}{dx^{11}}=100$$ Can ...
Holasoy Mas's user avatar
3 votes
1 answer
117 views

What is the meaning behind "Let" in this instance?

If we were given a few conditions on a complex $z$, say "The modulus of $z$ is $20$" and "the argument of $z$ is $\pi/3$", its clear that $z$ is unique. To find $z$ we might start ...
Nav Bhatthal's user avatar
  • 1,057
1 vote
1 answer
734 views

Family Members Birthday Dates all different, but our birthdays will fall on same day, even Leap Years. There is a total of 9 in this Birthday Club. .

I can compile a list if needed and post, but I noticed this over 50 years ago, My Father, My Brother and Myself our Birthdays fall on the same day of the week every year. Even Leap years, that does ...
David Rinkes Pastor David's user avatar
2 votes
0 answers
38 views

How do I find the constants in this differential equation that describes the time taken for a bubble to rise to the surface.

I'm trying to model the behaviour of bubbles in a water column as accurately as I'm able to, and I modelled a Differential equation using F = ma and a free body diagram. The equation of motion I ...
Movin Jayasinha's user avatar
0 votes
0 answers
54 views

Pull out constant from Laplace operator with rescale factor

Say I have a 2 dimensional cartesian laplace operator for a function $\Delta f = \frac{\partial^2 f}{\partial X^2} + \frac{\partial^2 f}{\partial Y^2}$ Part of the exercise is to scale the coordinates ...
reklem2's user avatar
14 votes
0 answers
944 views

Asymptotics of sequence of rational numbers

There is a simple sequence of rational numbers. It starts from $a_1=1$, and then $$ a_{n}=\begin{cases} a_{n-1} &\text{for even }n \\ a_{n-1}-\frac1n a_{\frac{n-1}2} &\text{for odd }...
AAK's user avatar
  • 716
1 vote
2 answers
59 views

real numbers, constant symbols in first order logic

I have this exercise in my book on first order logic: Let's work out a language for elementary trigonometry. To get you started, let us suggest that you start off with lots of constant symbols- one ...
user394334's user avatar
  • 1,262
0 votes
1 answer
126 views

Declaring constants in Sagemath

I want to find the Groebner base of a ideal,the ideal is generated by some polynomials with constant coefficients, but they do not have numerical values. ...
johnyy's user avatar
  • 31
6 votes
6 answers
495 views

Show that : $\frac{1}{\zeta(3)}<2C-1$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . It recall me the Faulhaber problem which relate the case $n=1,3$ with a square . ...
Ranger-of-trente-deux-glands's user avatar
8 votes
3 answers
478 views

Show by hand $\int_{0}^{\sqrt{5}}x^{x}dx>4$

It's a very Challenging question perhaps a kind of olympiad question : Prove that : $$\int_{0}^{\sqrt{5}}x^{x}dx>4$$ It's pretty sharp since the left hand side is almost $4.0005$ . I recall the ...
Ranger-of-trente-deux-glands's user avatar
2 votes
1 answer
75 views

Difficult limit $\lim_{x\to 0}f(x)=^?e$

Well I continue my previous question with : Define : $$f(x)=\left(\left|\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{2-t-n}\right)dt-x-1\right|\right)^{\frac{1}{x}}$$ Does we have : $$\lim_{x\to 0}f(x)\...
Ranger-of-trente-deux-glands's user avatar
2 votes
3 answers
128 views

What is $\lim_{x\to \infty}f(x)=\lim_{x\to \infty}\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{-t-n}\right)dt-x+1$

Working on the Gamma function I found the following: Let $$f(x)=\int_{0}^{x}\prod_{n=1}^{\infty}\left(1-e^{-t-n}\right)dt-x+1.$$ What is the limit $$\lim_{x\to \infty}f(x)=\,?$$ As possible guess it's ...
Ranger-of-trente-deux-glands's user avatar
2 votes
0 answers
128 views

Catalan's constant unexpected closeness

These days I usually wander around on WolframAlpha to experiment and discover many trivial but curious calculations or mathematical relations. Recently, I have randomly discovered a strange closeness ...
γ-stupid-like-irrationality's user avatar
0 votes
2 answers
105 views

Minimum of the Gamma/factorial function using dichotomy and Lagrange inversion theorem .

I want to evaluate the minimum of the Gamma's function via a kind of dichotomy . We have to start the value : $$\left(\frac{1}{\sqrt{5}}\right)!\simeq 0.8856$$ Solving $$x!=\left(\frac{1}{\sqrt{5}}\...
Ranger-of-trente-deux-glands's user avatar
1 vote
1 answer
105 views

Continued fraction inequality with well know constant as $\pi$ and golden ratio.

I found this inequality beautiful let me share it : Let : $$d=\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\...
Ranger-of-trente-deux-glands's user avatar
0 votes
2 answers
91 views

Conjecture about the minimum of the Gamma function

Problem/Conjecture: Let the function : $$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$ Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$ ...
Ranger-of-trente-deux-glands's user avatar
4 votes
0 answers
167 views

On the irrationality of $\zeta(\frac{3}{2})$

It is known that $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$Where $\zeta$ is riemann's zeta function. Usually people make $s$ an integer. But I thought of non integer values of $s$ and started with $s=...
Kamal Saleh's user avatar
  • 6,549
1 vote
0 answers
35 views

What's the arithmetic form of the value $k$ in this graph?

https://www.desmos.com/calculator/m1kaifseq0 in order to get the control points to line up properly, I had to use trial and error to determine the value for this strange constant $k$ at the bottom of ...
Maurdekye's user avatar
  • 327
2 votes
1 answer
69 views

About an odd constant in a longest arrangement problem

In my previous question about the longest chain of $n$-digit square numbers where last digit equal the first digit of next, the nice solution given by Misha Lavrov took me to consider the ratio of the ...
user967210's user avatar
2 votes
1 answer
175 views

Which number is greater A or B?

Let : $$I_k=\int_{0}^{1}\left(\prod_{n=1}^{k}\left(1+\arctan\left(\left(\frac{y}{4n^{2}}\right)\right)\right)\right)dy$$ And : $$h\left(x\right)=\int_{0}^{1}\left(\prod_{n=1}^{\operatorname{floor}\...
Ranger-of-trente-deux-glands's user avatar
0 votes
1 answer
93 views

Other closed forms of $\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)$

It is known that $$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\beta$$converges. I wonder if there are any other closed forms for this limit. At first I ...
Kamal Saleh's user avatar
  • 6,549
1 vote
3 answers
180 views

$\lim\limits_{x\to \infty}[f(x)-f(x-1)]\overset{?}{=}e$

Let : $$f\left(x\right)=\int_{0}^{\lfloor x\rfloor}\prod_{n=1}^{\lfloor x\rfloor}\frac{\left(y+2n\right)\ln\left(y+2n-1\right)}{\left(y+2n-1\right)\ln\left(y+2n\right)}dy$$ Conjecture: $$\lim_{x\to \...
Ranger-of-trente-deux-glands's user avatar
6 votes
1 answer
307 views

Euler-Mascheroni Constant in the Cosine Integral?

I came across this integral when doing calculus homework (Integration by parts) $$\int \frac{\cos x}x \, dx $$ It turned out in the end that there was a typo in the original question and this integral ...
ThreeMilks's user avatar
4 votes
0 answers
103 views

Prove that any continuous function $f:S_\Omega \rightarrow \mathbb{R}$ is eventually constant.

Let $\Omega$ be the first non-numerable ordinal number ( $\aleph_1$ is the first cardinal number greater than $\aleph_0$ when treated as an ordinal number is denoted by $\Omega$ ) and let $[0,\Omega)$ ...
Marco Roys's user avatar
0 votes
1 answer
125 views

How to find $C$ (improving a result due to user Clement.C)?

I improve the result see (Elegant) proof of : $x \log_2\frac{1}{x}+(1-x) \log_2\frac{1}{1-x} \geq 1- (1-\frac{x}{1-x})^2$ : What is the best constant $C$ such that $x\in(0,0.5]$: $\frac{1}{\ln\left(2\...
Ranger-of-trente-deux-glands's user avatar
4 votes
1 answer
159 views

Prove $f(z) = cz$ for all complex numbers and some $c$

Suppose $f$ is entire and $|f(z)|\geqslant |z|$ $\forall z \in \mathbb{C}$, prove there exists $c \in \mathbb{C} $ such that $f(z) = cz $ $\forall z \in \mathbb{C}$. I want to use Liouville’s theorem ...
Pegi's user avatar
  • 540
0 votes
0 answers
91 views

Finger's Constant

A fingers'constant is defined as to have as first decimal : $$1.2345...$$ I ask for the most emblematic example . For example : $$2-\prod_{k=1}^{\infty}\left(1-\frac{1}{\left(k+1\right)e^{k}}\right)$$ ...
Ranger-of-trente-deux-glands's user avatar
5 votes
2 answers
471 views

Show that $\int_{0}^{\infty}x^{-x}dx<\pi-\ln\pi$

I ask for an inequality which is a follow up of this question: Prove that $\int_0^\infty\frac1{x^x}\, dx<2$ : $$\int_{0}^{\infty}x^{-x}dx<\pi-\ln\pi$$ You can find a nice proof @RiverLi among ...
Ranger-of-trente-deux-glands's user avatar

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