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

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3
votes
1answer
105 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
1
vote
1answer
46 views

calculating the position of a given digit in a constant (e.g. $\pi$)

I'm aware that there are a lot BBP type formulas out there which extract the n-th digit of the observed constant. I'm asking for the reverse action, namely, is it possible to find the first ...
2
votes
3answers
378 views

How to rewrite $\frac{d}{d(x+c)}$? [closed]

I would like to know how to rewrite the following equations: $$ \frac{d (f(x))}{d(x+c)} =0\\ \frac{d^2 (f(x))}{d(x+c)^2} =0\\ $$ Here $x$ is a variable, $c$ is a constant and $f(x)$ is a function of ...
5
votes
3answers
41 views

Logs - Simplifying with arbitrary constant

I've tried simplifying my answer, which has a constant in it. I would like to know if I am on the right track: $$ \ln(y) = -{x^2\over 2y^2} + C $$ C can be considered as an Arbitrary Constant. From ...
2
votes
1answer
50 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
8
votes
1answer
92 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
1
vote
1answer
89 views

When is $\zeta(s)=0$?

At what real constant does $\zeta(s)=0$? Does that constant have any significance? Thank you very much for any help provided.
0
votes
1answer
24 views

What is the summatory function of the number of (not necessarily distinct) prime factors?

In the Math World article on Merten's Constant, a related constant $B_2$ is mentioned which "appears in the summatory function of the number of (not necessarily distinct) prime factors." I am very ...
3
votes
2answers
57 views

How is the Twin Primes Constant useful? What value does it provide over Brun's Constant?

The Twin Primes Constant is: $$\prod_{p > 2 \text{ and a prime }}\left(1 - \frac{1}{(p-1)^2}\right) = 0.6601618158\ldots$$ It appears that in this case $p$ does not have to be a prime. But if ...
4
votes
0answers
47 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
0
votes
0answers
28 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
0
votes
2answers
35 views

When do I have to respect the $C$ constant and when can I combine?

Question Verify that the given two-parameter family of functions is the general solution of the non-homogeneous differential equation on the indicated interval. $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ ...
3
votes
1answer
72 views

Mathematical importance of the golden ratio [duplicate]

I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with ...
-2
votes
1answer
93 views

How values of the constants are derived mathematically? [closed]

As said by Jan regarding constant value $\pi$ ,Imagine you have a circle and you are able to measure its circumference "c". Then, you can also find out what its diameter "d" is. When you divide ...
0
votes
1answer
72 views

What is the most intuitive explanation for euler's identity? [duplicate]

Is there any intuitive explanation for: $$e^{i\pi} + 1 = 0$$ About whether this question is a duplicate, what is asked for is not a proof but an explanation that helps with the not-so-intuitive ...
-1
votes
1answer
86 views

What is $1 \over \infty$ really? [closed]

Let's define infinity ($\infty$) as the number larger than any finite number. Also, let's define the infinitesimal constant ($\epsilon$) as the smallest number greater than zero. What is ...
1
vote
2answers
61 views

Online encyclopedia of maths and physics constants?

The OEIS is amazing, I used to spend hours creating sequences using arbitrary algorithms and look it up to see if it has an alternate definition and what's the meaning and the history of this ...
0
votes
0answers
12 views

How to find the constant of an expanded binomial expression?

Can somebody please explain how to find the Constant Term in an Expanded Binomial Expression? I have looked online and a lot of the explanations have confused me. The problem i'm currently on is ...
0
votes
3answers
162 views

Difference between variables, parameters and constants

I believe the following 4 questions I have, are all related to eachother. Question 1: Of course I've been using constants, variables and parameters for a long time, but I sometimes get confused ...
5
votes
0answers
70 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
1
vote
1answer
184 views

What is a “constant fraction” of a total?

What is it mean to say that some quantity is a "constant fraction" of another quantity?
0
votes
0answers
41 views

What is he easiest way to approximate γ as a decimal number?

What is the easiest way to give the numerical value of the Euler-Mascheroni constant? The mathmetical way to give that value? Thanks lot!
0
votes
1answer
33 views

Find A,B such that the given function is a solution to the given differential equation

The equation is: $x(t)=A\sin(t)+B\cos(t), \; x' - 3x = \frac{1}{2}\cos(t)$ I'm honestly just lost from where to start. I'd really appreciate any help. Answer: $A = \frac{1}{20}, \; B= ...
2
votes
0answers
47 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...
4
votes
1answer
121 views

Sequence with Prime Numbers

I was looking a question in a calculus book which used the following steps to show that following sequence has a limit (called Euler's constant $\gamma$): $$t_n = \sum_{i=1}^n\left(\frac{1}{n}\right) ...
1
vote
1answer
26 views

can a constant in a max be taken outside

A really silly question, but can I do: $$\max_x \left( c \cdot f(x) \right) = c \cdot \max_x f(x)$$ It seems that way, since I'm just interested in the maximum value of $f(x)$ which is not influenced ...
0
votes
3answers
49 views

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$? I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?
0
votes
1answer
20 views

$\sum_{n=2}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_{n+1}} = C$?

With $p_n$ prime, does the constant, $C$, exist and have a name? $$\sum_{n=1}^{\infty}[\log{(\log{(p_{n+1})})} - \log{(\log{(p_n)})}] - \frac{1}{p_n} = C$$ If not,how about a constant and function ...
3
votes
1answer
53 views

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
0
votes
0answers
114 views

Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. so for example 1/1^1 + 1/2^1 + 1/3^2 +1/4^2 + 1/5^4 + 1/6^2 etc... I also want to ...
1
vote
0answers
32 views

The probabilistic interpretation of ramanujan's constant $ e^{\pi\sqrt{163}}$

Some of the mathematical constant has an interesting probabilistic interpretation. For example, "$\large\pi$". Suppose two integers are chosen at random. What is the probability that they are ...
1
vote
2answers
80 views

Sum of a Normal and a Truncated Normal distribution

I have normal distribution $ N(\mu_1, \sigma_1)$ which shows the amount of demand in warehouse 1. The current amount of stock in the warehouse 1 is C. If the random demand is greater than C, it cannot ...
-2
votes
4answers
89 views

Why do texts frequently define $\mathbf {i}$?

Often when I see a formula containing $\mathbf {i}$, it will be accompanied by the definition $\mathbf {i^2 = -1}$. Why don't we just assume that most students of advanced math know what $\mathbf {i}$ ...
0
votes
1answer
43 views

What is the value of a + b + s + t

I'm confused as to how to approach this. I've tried to foil it out. $$x^3(3x-1)=a+bx+sx^2+tx^3$$ The equation above is true for all values of $x$, where $a$, $b$, $s$, and $t$ are constants. What ...
4
votes
1answer
56 views

Prime numbers making constant : 1.2527

Reading "Excursion in calculus" (Robert M. Young, 1992), exercice 13 on page 71 ask the reader to show there is a constant $c\approx 1.25$ such that $a_0=2^c$ $a_{n+1}=2^{a_n}$ $\forall n\; \lfloor ...
16
votes
4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
4
votes
0answers
389 views

MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper whose author has passed away. ...
1
vote
0answers
43 views

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation?

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation? $2$ is the only prime with $1$ one, the Fermat ...
0
votes
0answers
66 views

Lax Pairs and constant eigenvalues

Can someone tell me whether the following is true, and if so a hint the proof? If we have a Lax Pair $\dot{L} = [A,L]$ then the eigenvalues of $L$ are constants of the motion. (The opposite ...
2
votes
1answer
54 views

Terms that cannot be solved for a variable

Yesterday our analysis professor told us you cannot solve $$ y = e^x+2/(1+x^2) $$ for x, but you have the option to approximate this numerically. He did not prove that, he just noted it. I can't ...
2
votes
1answer
77 views

Does this have a name? (Regarding ways to calculate e)

Just wondering...came across this relationship regarding Euler's number in my math tinkerings, but I'm unaware if this particular relationship has a specific name or not: ...
1
vote
1answer
59 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
0
votes
1answer
82 views

How to prove this equality about e

I have the assumption that the following holds: $$\lim_{n \to \infty}\frac{1}{n^2} \cdot \sum_{i = 0}^n \left(1 - \frac 1n \right)^i = 1 - \frac{2}{e}.$$ However, I am totally not sure about it. How ...
2
votes
1answer
42 views

Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods?
3
votes
1answer
122 views

Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.

Consider $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. Bellow are series $(3)$ and $(6)$ that represent $\varphi$ $$ \begin{align*} \varphi &=\frac{1}{1}+\sum_{k=0}^{\infty}\cdots&(1)\\ ...
2
votes
1answer
72 views

What is the algebraic role of the mathematical constant $\gamma$?

Mathematical constants $\pi$, $e$, $i$ have a lot of algebraic roles. They appear as identity elements, idempotents, invariant elements etc against various operations and sets. This is illustrated by ...
1
vote
3answers
130 views

How to solve following limit

I've been struggeling a bit with the following limit: $\lim\limits_{x \to 0} \frac{a- \sqrt{a^2 - x^2}}{x^2}$ The solution is: If a < 0 then -$\infty$ . If a > 0 then $\frac{1}{2a}$ But I don't ...
0
votes
1answer
148 views

What's the value of tau?

I've seen $\tau$ on a title of a YouTube video and I need help knowing what the value is. I'm serious. I've never heard of the value. So, what is it? Also, is it rational or irrational (this part ...
1
vote
0answers
40 views

Linear second order PDE with constant coefficients

I am doing this mathematical problem $c*G_{vv}+ d*G_{u} + e*G_{v} +f*G=0$, where $c, d, e$ and $f$- are constant coefficients. I already know that this is second order PDE and we classify it by ...
0
votes
0answers
66 views

Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...