Questions tagged [constants]

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

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how to prove this identity $s = 2 \int_{- \pi}^{\pi} | \frac{\sin (t) - i}{(\sin (t) + i)^2} | dt = 2 K (- 1) = 2.62206...$

How can we prove this identity? Which, btw, Mathematica know how to simplify so it is missing some fundamental identity (related to the lemniscate constant.) \begin{equation} s = 2 \int_{- \pi}^{\pi}...
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1 vote
2 answers
175 views

Show the inequality $\frac{\sqrt{\pi}}{2}<\left(\pi-e\right)!$

Working a bit on About the inequality conjectured as $x!>\left(\arctan\left(\cosh\left(x\right)\right)\right)^{a}$ for $x>0$ and fixed $a$ I got the inequality: $$\frac{\sqrt{\pi}}{2}<\left(\...
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($C^1$ Function)False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f'$ is constant...

False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f(x)$ is constant. b) If $f$ is differentiable, then $f$ is of class $C^1$. In the case of true prove, in ...
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Correct timing of denoting an arbitrary constant of indefinite integral

$$\begin{align} A&:=\int{\arctan(x)\over 1+x^2}\,\mathrm{d}x\tag{1} \\ &=\int\arctan(x){\mathrm{d}\over\mathrm{d}x}\arctan(x)\,\mathrm{d}x\tag{2} \\ &=\arctan(x)\arctan(x)-\int{\arctan(x)\...
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59 views

How to show $2+\left(\frac{1}{2}\left(\sqrt{\frac{e^{e^{-1}}}{e^{-e^{-1}}}}+\sqrt{\frac{e^{-e^{-1}}}{e^{e^{-1}}}}\right)\right)^{2}<\pi$?

Problem : $$2+\left(\frac{1}{2}\left(\sqrt{\frac{e^{e^{-1}}}{e^{-e^{-1}}}}+\sqrt{\frac{e^{-e^{-1}}}{e^{e^{-1}}}}\right)\right)^{2}<\pi$$ Some related work : You can find some material here Showing $...
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1 vote
2 answers
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Proportionality constant

I’ve typed the question, in whose context my doubt is, and it’s answer at the end. Please note that I do not require the solution as I’ve already understood how to find the answer via the given as ...
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3 votes
1 answer
29 views

Need Help with an Integral Formula for Alternating Sum of Reciprocals of Logarithms

I was curious as to the limit of $\sum_{2}^{\infty} \frac{(-1)^n}{\ln(n)}$, and eventually found the sequence of its digits on the OEIS, sequence A099769. On there, the expression: $$\frac{1}{2\ln(2)} ...
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Find $C$ such that $\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^{2}+\beta x_{i+1}^{2}}\geq \frac{\sum_{i=1}^{n}x_i}{C}$ is true

Problem Let $x_i>0$ and $n\geq 3$ then find the best constant which is a natural number $C=\alpha+\beta$ with $\alpha,\beta >0$ natural numbers such that $$\sum_{i=1}^{n}\frac{x_i^3}{\alpha x_i^...
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Why do these identities give the MRB constant for "2n"?

After a lot of searching, I've came across a couple of formulas that give the MRB constant for all values of 2n that I use. They are $$\text{CMRB}=\frac{1}{2}+\int_1^{i \infty } -(\csc(\pi t) ) \...
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2 votes
1 answer
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Am I allowed to define $e$ through algebraic means using this limit?

By using the formal definition of a derivative $$f'(x) = \lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ one can get $$\frac{d}{dx}[e^x] = e^x \times \lim \limits_{h \to 0} \frac{e^h-1}{h}$$ which ...
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2 votes
3 answers
40 views

What rationale am I missing with this simple boundary value problem?

Consider $$u_{xy} = xy,\ u(0,\ y) = 0,\ u_x(x,\ 0) = 0$$ Antidifferentiating the PDE with respect to $y$ yields $u_x = \frac{xy^2}{2} + a(x)$, and antidifferentiating this equation with respect to $x$ ...
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-1 votes
1 answer
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Approximation of pi : Show the inequality without a calculator .

Hi it's an inequality found by chance using an integral analogue of the Kantorovitch inequality . The problem : $$300+\left(\int_{0}^{10}2-x^{-x}dx\right)\cdot\left(1+e^{-1}\right)^{-2}\cdot4\left(e^{-...
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1 answer
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Prove that sum of velocity and squares of position is a constant using the function given

An object moves along a line modeled by the x-axis. Its position (i.e. x-coordinate) after $t$ seconds is given by \begin{equation} x(t)=a\sin t + b \cos t \end{equation} where $a$ and $b$ are ...
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-1 votes
1 answer
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Integral of a rational function with varying exponents [closed]

Find the exact value of the following integral, where $r,s \in \mathbb{R} $ and $ 0<r<s$. $$\int_{0}^{\infty}\frac{x^{r-1}}{1+x^s}dx$$ I have looked 3 cases: Case 1: $0<r,s<1$ Case 2: $0&...
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1 vote
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Is this old news? $\sum_{i=1}^n \frac{i!}{p_i\#} \approx 1.240053652689\dots$

This is a soft question as it arises out of my curiosity alone. I noticed that as $n$ increases, $\frac{n!}{p_n\#}$ decreases in magnitude much faster than $\frac{1}{p_n}$, and I wondered if the sum $$...
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9 votes
1 answer
207 views

Is there a closed form of the Laplace Limit Constant: $x$ such that $\frac{xe^{\sqrt{x^2+1}}}{\sqrt{x^2+1}+1}=1$ using library functions?

The Laplace Limit Constant $\lambda$ is well know constant which is the $y$ value of the global extrema of: $$x\,\text{sech}(x):$$ Therefore: $$x=\max(x\,\text{sech}(x))=-\min(x\,\text{sech}(x))=1....
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1 vote
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What does it mean to say "the inequality is tight up to constant factors"?

On the Wikipedia page for Pinsker's inequality, it states "the inequality is tight up to constant factors". $$ \delta(P, Q) \leq \sqrt{\frac 1 2 D_\text{KL}(P||Q)} $$ What does this mean? ...
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1 answer
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Mean value theorem question - proving F constant - answer check

The question: Given $f$ continuity at $[a,b]$ and derivative at $(a,b)$. It is known $f'(x)=0$ for each x belongs to $(a,b)$. prove $f$ is constant. My Answer: Need to prove $f(x)=k$ for each X ...
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0 votes
1 answer
39 views

PDEs With Partial Derivatives W.R.T. a Single Variable

Is it always correct to solve partial differential equations as though they were ordinary differential equations if the partial derivatives are only taken with respect to a single variable, even if ...
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6 votes
0 answers
218 views

Find extrema of $y=?(x)-x$ with the Minkowski Question Mark function

The Goal: is to figure out the global extrema of the Minkowski Question Mark function $?(x)$. Here is the graph of: $$?(x)-x:$$ The $y$ value of the global maximum was found by systematically ...
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0 votes
0 answers
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Solving a PDE Via the Similarity Method Versus Another Method

The PDE is $$u_{xx} + 2u_{tt} = 0$$ I imagine that the solution will be $u(v(x,\ t))$, where $v(x,\ t) = \frac{t}{x}$. So plugging this form into the PDE and using the multivariable chain rule yields ...
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0 votes
1 answer
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How to find constants in a function if the function and its derivative is given?

If the function is : $$f(x) = (2x-b)^a$$ And it's derivative is: $$f'(x) = 24x^2-24x+6$$ Then find the value of $a$ and $b$ I tried by calculating the derivative of $f(x)$ which comes out to be: $$f'(...
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1 vote
2 answers
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What is $\lim\limits_{n\to\infty} (\text j_{x,x}-\text y_{x,x})$ with the BesselYZero and BesselJZero function?

This question is similar to: Conjecture: $$\lim\limits_{x\to\infty}\operatorname{Re}\text W_x(x)\mathop=\limits^?-\ln(2\pi)$$ I have come across BesselJZero and BesselYZero function as a form of “...
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1 vote
1 answer
158 views

An upper bound for the function $x!$ using some well-know constant as $e$ or $\pi$

Problem : Define : $$f(x)=\left(e^{\frac{3x\left(\pi^{x}-e^{x}\right)}{\pi^{x}+e^{x}}}-e^{\sqrt{2x}-3}\right)^{\frac{3}{\pi}}$$ Let $x>\frac{1}{10}$ then prove or disprove that : $$j(x)=f\left(x\...
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0 votes
0 answers
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Failure to Lump Constants in Laplace's Equation

Consider a $2D$ version of Laplace's Equation, for example, $\triangle u = 0, u(x, 0) = 3, u(x, 1) = 3, u(0, y) = 0, u_x (1, y) = 0$. Separation of Variables leads to the general solution $u = (Ae^{-\...
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3 votes
4 answers
107 views

$\int_0^y e^x dx = e$, solve for y

$\int_0^y e^x dx = e$ What is $y$ here? The definite integral from $0$ to $1$ is $e - 1$. But what number must be this integral's upper limit in order to produce an area under the curve of $e$? ...
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  • 375
-1 votes
1 answer
55 views

Show that the function $f(x)=\left(\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\right)-\frac{1}{x}+\left(\frac{e}{2}-\frac{x}{e^{x}}\right)-x$ is negative

Hi I proposed a similar question some days ago and I cannot find the answer .Now the problem : Let $0<x$ then we have : $$f(x)=\left(\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\right)-\frac{1}{x}+\left(\...
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0 votes
1 answer
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Constant function if the derivative is null

Let $f$ be a function defined on $(a,b)$. I know that $$f'(x)=0,\ \forall x\in(a,b)\iff f\, \text{constant in}\, (a,b)$$ I know that if the domain of the function is not an interval then this result ...
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2 votes
1 answer
122 views

Show without a calculator : $e^{-\gamma}<\omega$

Hi I think this question is new : Problem : Show that : $$e^{-\gamma}<\omega$$ Where we have the Euler's number and constant and omega constant wich is the value taking at $x=1$ of the Lambert's ...
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8 votes
5 answers
489 views

Showing $\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$ without a calculator

Show that: $$\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$$ I have tried power series of exponential around $0$ wich is : $$e^x=1+x+\frac{x^2}{2}+O(x^3)$$ We can ...
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2 votes
0 answers
75 views

About the minimum of the Gamma function on $(0,1)$

Problem : Denotes by $x_{min}=k$ the minimum (not the abscissa) of the Gamma function $x!$ on $(0,1)$ then prove or disprove that : $$\left(e^{-\frac{k^{2}}{C^2}}\right)!>k$$ Where $C=-1+\frac{1}{\...
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1 vote
1 answer
45 views

Show that $h(e)<f(e)$ where e is the exponential

Problem : Define for $a=0.5$ and $x=e$: $$f(x)=\left(1+\frac{1}{x+a}\left(\frac{4\left(x+a\right)^{2}}{\left(x+a+2\right)^{2}}-1\right)\right)\left(x+a-\frac{2\left(x+a\right)}{\left(x+a+2\right)}\...
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2 votes
2 answers
110 views

Show that $\ln(\sin(-20)+21)>3$ by hand

Hi I hope this problem is new : Show that : $$\ln(\sin(-20)+21)>3$$ I have tried the power series of $\ln(x)$ and $\sin(x)$ without success because it's becomes hard by hand . You can also find the ...
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1 vote
0 answers
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Second derivatives test - Finding unknown constants

With the shape given by equation $Az = Bx^m + Cy^n − Dx − Ey + 14$ where $x$, $y$, and $z$ are measured in meters. If you are standing at location $(10, 30, 40)$, by performing second derivatives test,...
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0 votes
1 answer
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Find out constants$~a,b,c,d~$such that$~\lim_{x\to0}\frac{\sin^{}\left(3x\right)-\left(ax^{2}+bx+c\right)}{x^{3}}=d~$is satisfied

$$\left(a,b,c,d:=\text{constants}\right)~~\wedge~~\left(d\neq0\right)$$ I want to find out the formula(s)or value(s)of the above constants which satisfy the following equation. $$\lim_{x\to0}\frac{\...
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2 votes
1 answer
96 views

On the evaluation of $\sum\limits_3^\infty \frac1{\ln\Gamma(n)}$

Motivation: This question will be inspired from: Evaluation of $\sum\limits_{n=1}^\infty \frac 1{\text G(n)}≈ 3+\frac{\,_0\rm F_2(2,3;1)}2 $ with the Barnes G function? and Evaluating $\sum\limits_{x=...
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3 votes
0 answers
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Pulling Some Threads of the 2nd Order PDE Technique

I have some conceptual questions regarding a solution technique for second order linear PDEs. The example I have been considering is $u_{xx} + 2u_{xy} + u_{yy} = 0$. The technique is to use the guess ...
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2 votes
0 answers
189 views

Evaluation of $\sum\limits_{n=1}^\infty \frac{(-1)^n\text{Ei}(n)}{n!}$

Motivation: This sum came up in a sum of a central gamma function problem: Evaluation $$\sum_{-\infty}^{-1} \Gamma(n,n)= \pi\left(\frac1e-1\right)i+ \sum_{n=1}^\infty \frac{(-1)^n \text{Ei}(n)}{n!}+\...
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0 votes
0 answers
17 views

Integral Constant Increasing with Equation Output

I am working on an analytical solution for a numerical model, I have integrated the formula dT2/dt = FCp(T1-T2)/M2*Cp Where T1 = Temperature in First Reactor T2 = Temperature in Second Reactor F = ...
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2 votes
0 answers
134 views

Evaluation of $\sum\limits_{n=1}^\infty \frac 1{\text G(n)}≈ 3+\frac{\,_0\rm F_2(2,3;1)}2 $ with the Barnes G function?

I thought this problem would be arbitrary, but is really easier to calculate than the inspiration for this question: Evaluating $\sum\limits_{x=2}^\infty \dfrac{1}{!x}$ in exact form. Now our ...
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4 votes
0 answers
138 views

What determines how to treat single variable PDEs and thus their constants of integration?

When solving a first-order ODE (perhaps there is also a way to extend this to a higher order ODE) for $y(x)$, it is possible to shift perspective and consider $x$ to be a function of $y$ by ...
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-1 votes
2 answers
50 views

Constant after integration [duplicate]

What is the true significance of the constant "c" that we add after we integrate a curve without applying limits?
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0 votes
1 answer
134 views

Finding the constant of a joint probabiity distribution

Let $A$ and $B$ be random variables such that, for some $\theta>0$, $P(A=x,B=y)$ = $\theta$$\frac{2x+y}{x!y!}$$(0.33^{x+y})$ for $x=0,1,2,3,…$ and $y=0,1,2,3,….$ How do I find the value of $\theta$?...
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0 votes
0 answers
167 views

Does this Auxiliary Fresnel Sum=$\frac1{2\sqrt2\pi}\int \limits_0^\infty \frac{\vartheta_3\left(e^{-\frac{\pi x}2}\right)\sqrt x}{x^2+1}dx +\frac14 $?

$$\large{\text{Motivation:}}$$ Here is a related Fresnel Integral sum for a seventh in a series of a sum of just a single function: On $$\mathrm{\sum\limits_{n=0}^\infty \left(C(n)-\frac{\sqrt\pi}{2\...
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3 votes
0 answers
77 views

How to solve $x=\lim\limits_{t\to0} Q^{-1}(t,t)\implies \text{Ei}(-x)=-1\implies Γ(0,x)=1$?

Based on: Conjecture: $$\lim\limits_{x\to\infty}\operatorname{Re}\text W_x(x)\mathop=\limits^?-\ln(2\pi)$$ and On completing the solution for $$\int_0^1 Q^{-1}(x,x) dx$$ and other constants. Here ...
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0 votes
1 answer
94 views

Show that $f(x)<1$ for a special $x$

Let define the function : $$f\left(x\right)=\frac{2}{x\left(\tanh\left(xe^{-1}\right)+1\right)}$$ Show that : $$f\left(\frac{1+\sqrt{3}}{2}\right)<1$$ Some facts : $$\tanh(x)=\sum_{n=1}^{\infty}\...
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0 votes
0 answers
61 views

Does this continued fraction have a name? [duplicate]

$1/(2+1/(3+1/(5+...) = [0; 2,3,5,...p_{\infty}] \approx 0.432332$ Does this constant have a name? What is it called? It does appear to converge from my initial calculations, and I'm surprised that I ...
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1 vote
1 answer
38 views

derivative with respect to constant (Lagrange multiplier) .

I cannot understand how if a Lagrange multiplier is a scalar (meaning it is a constant value) that you can take a partial derivative of a Lagrangian function with respect to a constant (the Lagrange ...
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6 votes
2 answers
212 views

Computing zillions of digits of the "derangement constant"

This is a sort of inspired sequel to the following question: Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form. where the question is the discussion of the "$e$-like constant" ...
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0 votes
0 answers
42 views

Can I lump constant functions of n in this way?

I am trying to solve a separation of variables PDE problem using Complex Fourier Series to apply the final boundary condition. So far, I have the solution $u = \sum_{-\infty}^\infty (A_n r^{-n} + B_n ...
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