Questions tagged [constants]

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

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21 views

Prove that $\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$

$$\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$$ Where $\operatorname{li}(x)$ is the Logarithmic integral function $\Gamma(x)$ is the Gamma function $\gamma$ ...
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2answers
61 views

Prove that $\sum_{k=1}^{\infty}\frac{k^2}{e^k}< \sum_{k=1}^{\infty}\frac{k}{2^k}$

Prove that : $$\sum_{k=1}^{\infty}\frac{k^2}{e^k}< \sum_{k=1}^{\infty}\frac{k}{2^k}$$ Without calculating the value of these series . The partial sum formula are : $$\sum_{k=1}^n \...
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226 views

Swinging factorial and swinging constant

The Swinging factorial $n≀$ defined as $$n≀=\frac{n!}{\left\lfloor{n/2}\right\rfloor!^2}$$ is relatively common and I found some results on Google. But when $$\sum_{n=0}^{\infty}\frac{1}{n≀}$$is ...
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52 views

Prove that $2$ is the best constant

It's related to my own question Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$ Let $x\geq 5$ be a real number then we have : $$\frac{x^2+1}{x+1}\Bigg(\frac{x^{\frac{x}{x+1}...
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18 views

One way to solve $\int_{0}^{\infty}3\Big(\frac{e^{-x^3}}{x+1}+\frac{xe^{-x^3}}{x^3+1}-\frac{e^{-x^3}}{x^3+1}\Big)dx=G$

It's a simple question we have : $$\int_{0}^{\infty}3\Big(\frac{e^{-x^3}}{x+1}+\frac{xe^{-x^3}}{x^3+1}-\frac{e^{-x^3}}{x^3+1}\Big)dx=G$$ Where $G$ is the Gompertz constant It has a simple ...
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2answers
95 views

Finding constant for orthonormality

Assume the problem \begin{align} f''(x) &= \lambda f(x) \\ f'(0) &= f(1) = 0 \\ \end{align} with solution: $$ \phi_n(x) = c \cdot\cos\left( \frac{(2n-1)\pi}{2}x\right), \quad n = 1,2,\dots $...
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10 views

Improvement of the convergence to the Dottie number?

Hi I study the following sequence : Let $0<a_0<1$ then the following sequence converges : $$a_{n+1}=\frac{\cos^{\alpha}(a_n)}{a_n^{\alpha-1}}$$ Where $0<\alpha<1$ And :$...
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57 views

What constants make the integral equal to $1$?

What constants $a_1,a_2, \cdot \cdot \cdot, a_n \in \Bbb N $ make the following integral equal to $1$? $$ \int_0^1 \frac{a_1e^{\frac{1}{\log(x)}}}{x\log^k(x)}+ \frac{a_2e^{\frac{1}{\log(x)}}}{x\log^{...
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29 views

Find the constants $a,b,c$ to satisfy the re-writing of a function in another form. [closed]

Given $f(x) = 3x^2 - 4x + 5$ what are the constants a, b, c if $f(x)$ was written in the form: $a(x-b)^2 + c$
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14 views

Verhulst model and Lipschitz dependancy

I have a differential equation as follow which is Verhulst model: $$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$ So I wanted to see just if there is a solution to this equation and if it is ...
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23 views

Prove that $\frac{3}{50}<\int_{0}^{1}\exp\Big(-\operatorname{W^2(x)}\Big)\operatorname{W^e(x)}dx$

At the beginning I was thinking to the Laplace transform of the Lambert's function as there is no easy to way express this I propose this similar problem : $$\frac{3}{50}<\int_{0}^{1}\exp\Big(-\...
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50 views

Let $f$ be an entire function such that $f(z) = f(1/z)$ for all $z \in \mathbb{C} \setminus \{0\}$. Show that $f(z)$ is constant.

Let $f$ be an entire function such that $f(z) = f(1/z)$ for all $z \in \mathbb{C} \setminus \{0\}$. Show that $f(z)$ is constant. I understand Liouville's Theorem could come into play here but unsure ...
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36 views

Proof that $n$-th root of $n$-th partial convergent's denominator almost always tends to $e^{\frac{\pi^2}{12\ln{2}}}$

While reading Steven Finch's book Mathematical constants (I believe), I once came across and wrote down the following theorem: For almost all real numbers $x$, if $\frac{P_n}{Q_n}$ is the $n^{\text{th}...
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247 views

Hard problem : Prove that $\Gamma\Big(\frac{1}{2}\Gamma\Big(\frac{1}{2}\Gamma\Big(\frac{1}{2}\Big)\Big)\Big)<\frac{\pi^2}{6}$

It's a problem that I can't solve it's : $$\Gamma\Big(\frac{1}{2}\Gamma\Big(\frac{1}{2}\Gamma\Big(\frac{1}{2}\Big)\Big)\Big)<\frac{\pi^2}{6}$$ You have the difference here. What I know ...
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2answers
40 views

Total number of mathematical constants [closed]

Can we know how many interesting constants exist in all like $\pi$ and $e$?
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2answers
131 views

Find a closed form to the solution of $\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-x}}}}}=x$

Hi I try to solve the following nested radical : $$\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-x}}}}}=x$$ Miraculously the related polynomials is a quintic .More precisely : $$ x^5 - x^4 - 4 x^3 + 3 x^...
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1answer
24 views

Finding the Values of a Constant For Which A Curve has Local Maximum and Minimum Values

Use calculus to find the values of the constant $c$ for which the curve has local maximum and local minimum points. $g(x) = 4x^3 +cx^2 +10x$. Show that the graph always has one inflection point for ...
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2answers
24 views

Distributing 2 times k equals k?

in my current discrete mathmatics course I have this calculation at the end of a proof: $$\frac {k(k+1)+2(k+1)}2=\frac {(k+1)(k+2)}2$$ enter image description here I dont under stand why 2*k ends ...
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15 views

An example of nested radical and power tower .$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$

I want to share with you some of my last work: $$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$$ It's easy to solve using logarithm but I would like ...
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24 views

Almost integer with nested radicals and power tower .

playing with power tower and nested radicals I get : Prove that Let $a_1=\sqrt{2}$ ,$a_2=\sqrt{2}^{\sqrt{2}}$,$a_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}$,$a_4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}...
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40 views

Checking Answer to Homogeneous ODE System

The system is $$\begin{cases}x' = 4x - 13y \\ y' = x\end{cases}.$$ When I solve it on paper by finding the complex eigenvalues and eigenvectors, applying Euler's formula, and lumping constants, I get ...
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1answer
62 views

Little integral $\int_{0}^{\infty}\ln\Big(\frac{x^2-2x+1}{x^2+2x+1}\Big)e^{-x}dx$ related to the Gompertz constant

It's a little result related to the Gompertz constant We have : $$\int_{0}^{\infty}\ln\Big(\frac{x^2-2x+1}{x^2+2x+1}\Big)e^{-x}dx=-2G-2\frac{\operatorname{Ei(1)}}{e}$$ Where $G$ is the Gompertz ...
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25 views

Algebra point of view on Tribonacci constant with nested radical

Trying to find an expression in term of nested radical for the tribonacci constant we get a messy result starting with : $$S=\sqrt{1+S+\frac{1}{S}}$$ I have tried another expression like : $$S=\...
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24 views

Alternating Prime Sum

How can one evaluate this sum: $\sum_{n=1}^\infty (-1)^{n+1}/{p_n}$? I've tried it for myself but didn't get any satifying answer although it does converge. I also can't find this sum anywhere else ...
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2answers
42 views

Is the number $1.201~943\dots$ of any significance?

I ran the function $(x^2+y^2)^z=z$, where $z$ is a constant. The function produced a circle, and the $z$ value where the radius of the circle turned out to be the largest was $e$. At that point, the ...
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48 views

What about of the irrationality and transcendence of expressions involving the omega constant, and/or $\pi$ and $e$?

I know that there are some open problems concerning the irrationality and trancescende of certain combinations (suitable expressions as sums/differences, products/quotients and exponentiations/...
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115 views

reference request: Plouffe's Lambert-type series for $\zeta(2n+1)$

According to Wikipedia, Plouffe gives the series $$\begin{align} \zeta(5)&=\frac1{294}\pi^5-\frac{72}{35}\sum_{n\ge1}\frac1{n^5(e^{2\pi n}-1)}-\frac2{35}\sum_{n\ge1}\frac1{n^5(e^{2\pi n}+1)}\\ &...
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1answer
31 views

Harmonic oscillator constants, general solution.

While doing some calculations I have stumbled into some step, that I am unable to justify. The question is regarding harmonic oscillator and is as follows: Given simple harmonic oscillator we have ...
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1answer
52 views

Generalizing a Fundamental Matrix

In this video (final result at 8:54), the professor explains that the most general fundamental matrix of a system of linear ordinary differential equations is given by $XC$, where $X$ is any ...
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1answer
44 views

Characterize all the continuous functions that satisfy the following condition

Characterize all the continuous functions $f$ in $[a,b]$ that satisfy $$\int_a^b f(x)\,\varphi(x)\,dx = 0$$ $\forall\,\varphi(x)$ continuous in $[a,b]$ such that $\int_a^b \varphi(x)\,dx = 0$. I ...
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1answer
89 views

Sine Wave With Alternating Wavelength

Please Read Everything Fully and Carefully Before Responding!! I'm trying to formulate parametric equations for a sine wave where the wavelength grows by a constant on alternate sides, in this case, ...
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1answer
109 views

Does the constant 4.018 exists?

Let $A$ be the set $\{a_1,a_2,\ldots,a_n\},$ for each $i, a_i $is prime number of the form $3j^2+2, j \geq 0 $ let $B$ be the set $\{b_1,b_2,\ldots,b_n\}$, for each $i, 3b_i^2+2$ is prime number,$ ...
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522 views

A magnificent series for $\pi-333/106$

Stated here without proof is the magnificent series $$\frac{48}{371} \sum_{k=0}^\infty \frac{118720 k^2+762311 k+1409424}{(4 k+9) (4 k+11) (4 k+13) (4 k+15) (4 k+17) (4 k+19) (4 k+21) (4 k+23)} \\=\pi-...
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42 views

Prove function is constant [duplicate]

Let $f: \mathbb R \to \mathbb R$ be a real function, which is continuous in 0. There exists one $a \in \mathbb R \backslash\{-1, 1\}$ with $f(x) = f(ax)$ $~~~~(x \in \mathbb R)$. Prove that $f$ is ...
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60 views

Is there any reason that $\arctan(e)\approx e-\frac32$ or is it just pure number luck?

Is there an informative interpretation to why $$\arctan(e)\approx e-\frac32$$ They differ by $1.07 \times 10^{-6}$.
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5answers
116 views

What are some large or small mathematical constants?

I understand this question is a bit vague, but I would like to know about notable mathematical constants that are large or small, and I clarify what I mean now: Notable, as in not a product, sum, ...
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87 views

The structure of numbers that are not “Khinchin random.”

Khinchin's Continued Fraction Theorem: For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n(r) = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ ...
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1answer
34 views

At what rate is the area of the surface of the plate increasing when its diameter is 6 meters?

A circular metal plate is heated so that is diameter is increasing at a constant rate of of $0.005\frac{m}{s}$. At what rate is the area of the surface of the plate increasing when its diameter is $6$ ...
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1answer
81 views

Gosper's identity for the golden ratio: $\frac{2^{2/5}\sqrt{5} \, \Gamma(1/5)^4}{\Gamma(1/10)^2 \,\Gamma(3/10)^2} = \phi$

Towards the end of a talk by Knuth (one of his Christmas talks, maybe the one from 2017), he mentioned in passing the following identity communicated to him by Bill Gosper (without proof, IIRC): $$\...
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60 views

About $\sqrt[k]{l + \sqrt[k]{l + \sqrt[k]{l + …}}} $ asymptotics

Consider simple nested radicals More precisely Let $$ K > 1 , 1 \leq l $$ $$X(j,K) = X_\infty(j,K)$$ $$X_0(j,K) = a(j,K)$$ $$X_n(j,K) = \sqrt[k]{j + X_{n-1}(j,K)}$$ $$Y(j,K) = \frac{j + X_{\...
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1answer
72 views

Fraction mapped to sum and geometric mean approaches 2.1479

Map a positive fraction $\frac{a}{b}$ to $\frac{a + b}{\sqrt{ab}}$. Repeating seems to map every starting fraction to a number close to $\xi = 2.1479$: $$ \begin{array}{ccccc} \frac{1}{4} & \frac{...
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1answer
37 views

Entire function which is a constant

Give f is entire, I have to show if $$\lim_{z\to\infty}\frac{\text{Re }f(z)}{z}=0$$, then $f$ is bounded. I've proved if $\lim_{z\to\infty}\frac{f(z)}z=0 $ then $f$ is constant by constructing new ...
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1answer
24 views

smallest perimeter of compound shape

If you have a compound shape made of three unique squares with fixed sizes, what is the smallest possible perimeter for that shape? assuming no overlaps.
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1answer
40 views

Solving a Riccati ODE Twice

$y' - \frac{1}{t}y = y^2 - \frac{3}{t^2}, y_p = \frac{1}{t}$ Method 1 To obtain a Bernoulli ODE, we plug $y = \frac{1}{t} + u$ into the Riccati ODE, yielding $u' - \frac{3}{t}u = u^2$. To obtain a ...
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2answers
70 views

Given three non-negative numbers $a,b,c$. Prove that $\frac{a+b+c}{k}\geqq\sum\limits_{cyc}\frac{a-b}{b+ k}$ for $k= constant$ so that $k> 0$ .

Given three non-negative numbers $a, b, c$. Prove that $$\frac{a+ b+ c}{k}\geqq \frac{a- b}{b+ k}+ \frac{b- c}{c+ k}+ \frac{c- a}{a+ k}$$ for $k= constant$ so that $k> 0$ . For $k= 2$, we can ...
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37 views

Given three positive numbers $x,y,z$ so that $xyz=1, xy+yz+zx=5$. Find the maximum value $x^{c}+ y^{c}+ z^{c}$ for $c\geqq -1$ .

Given three positive numbers $x, y, z$ so that $xyz= 1, xy+ yz+ zx= 5$. find the maximum value $$x^{c}+ y^{c}+ z^{c}$$ for $c\geqq -1$ . I think that the equality occurs at $x= y= 2, z= 1\div 4$ ...
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2answers
117 views

Infinite product $\Gamma(\tfrac14)=\mathrm{A}^3e^{-\mathrm{G}/\pi}2^{1/6}\sqrt{\pi}\prod_{k\ge1}\left(1-\frac1{2k}\right)^{(-1)^k k}$

I saw the following infinite product on Wikipedia: $$\Gamma\left(\tfrac14\right)=\mathrm{A}^3e^{-\mathrm{G}/\pi}2^{1/6}\sqrt{\pi}\prod_{k\ge1}\left(1-\frac1{2k}\right)^{(-1)^k k}\tag{1}$$ where $\...
3
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3answers
91 views

Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$

I recently saw on this site, the identity $$\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$$ which I instantly wanted to prove. I know that I can "reduce" the problem to the evaluation of $\...
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24 views

An constant I found whilst messing with movement on a 2D plane and powers of 2.

Let's say I have a system where I start at an origin and move to the right once and am now facing right. After that, I move either left or right according to these rules: If I can turn right without ...
3
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1answer
94 views

Prove that $e$ is irrational

Could you please verify whether my attempt is fine or contains logical gaps/errors? Thank you so much! Lemma: $0<e-\sum_{k=0}^{n} \frac{1}{k !}<\frac{1}{n n !}$ for all $n \in \mathbb N ^+$. ...

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