# Questions tagged [constants]

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

378 questions
Filter by
Sorted by
Tagged with
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 $\gamma$ ...
61 views

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 ...
95 views

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: $a(x-b)^2 + c$
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 ...
23 views

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 ...
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 ...
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 ...
24 views

522 views

60 views

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 ...
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.
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 ...
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 ...
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 $x= y= 2, z= 1\div 4$ ...
117 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 $n \in \mathbb N ^+$. ...