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

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-2
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0answers
88 views

Looking for a constant from complex analysis, ( and maybe other interesting constants) [closed]

Looking for a constant that is in complex analysis, it is about ratio of two types of sets in complex analysis. I have tried googling, wikipedia to no avail. I think I read about a constant ratios of ...
9
votes
0answers
130 views

Are my calculations of a new constant similar to Mill's constant based on $\lfloor A^{2^{n}}\rfloor$ and Bertrand's postulate correct?

As Wikipedia explains in number theory, Mills' constant is defined as: "The smallest positive real number $A$ such that the floor function of the double exponential function $\lfloor A^{3^{n}}\...
16
votes
1answer
2k views

Why can't I find anyone who has discovered the (irrational) constant 1.29128…? [closed]

The constant is exactly $\sum_{n=1}^∞\frac{1}{n^n}$. Why does it seem that no one has written about it? Did I not search well enough? If so, what is the name for it? If not, it is not sufficiently "...
6
votes
1answer
98 views

How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$ \int_{0}^{\infty}\frac{1}{\Gamma(x)}dx $$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the Fransén-...
1
vote
2answers
31 views

Show that an entire function is a constant [duplicate]

Let $f$ be entire and suppose there is a constant $M>0$ such that $|f(z)|>M$ for all $z \in \mathbb C$. Prove that $f$ is constant. I think this has something to do with Liouville's theorem but ...
0
votes
1answer
74 views

General solution of the differential equation: y' cot x + y = 2

I have to find the general solution of the differential equation:$ y$' $cot$ $x$ + $y$ = $2$. And determine the integration constant using the initial condition $y$(0) = $1$. Additionally presenting ...
1
vote
0answers
64 views

Ways to determine $\pi$ [duplicate]

I have read that it is possible to determine the value of a single digit, say the 874th of $\pi$. I know that it is a trascentental number, how is that possible? How many ways are there to determine ...
3
votes
1answer
89 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
7
votes
1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
1
vote
1answer
34 views

Family functions of derivatives and extremum values. [closed]

Explore the family of functions $ f(x) = x^3 + kx + 1 $ where $ k $ is a real constant. How many and what type of local extrema are there? Your answer should depend on the value of $ k $, that is, ...
0
votes
2answers
76 views

Why square a constant when determining variance of a random variable?

If I want to calculate the sample variance such as below: Which becomes: $\left(\frac{1}{n}\right)^2 \cdot n(\sigma^2)= \frac{\sigma^2}{n} $... My question is WHY does it become $$\left(\frac{1}{n}...
1
vote
0answers
43 views

Why do we care about the Champernowne constant?

I was browsing code golf and I came across this challenge: http://codegolf.stackexchange.com/questions/68685/the-rien-number It caught my interest and I wanted to learn a bit more about the ...
2
votes
3answers
52 views

Constant functions periodic?

I dont understand the meaning of this line in my book - " $\sin^2x + \cos^2x$ is periodic but the fundamental period is not defined. " Why is the period not defined? $F(x)$ is $1$ here so it is a ...
0
votes
1answer
32 views

Problem with constants in Solving first order differential equation with perturbation

$$y'+\lambda \ y^4 +y=0$$ Where $\lambda$ is very small and $y(1)=-0.5$. I've tried to solve it by Substituting: $y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots$ but I had problems with the integration ...
0
votes
1answer
24 views

Skipping integration constants

Here is simple decay equation – its initial conditions are being derived: $$ m(t) = Ce^{-kt}\\ m_0 = C^{-kt_0}\text{, which gives}\\ C = m_0e^{kt_0}\\ \text{After inserting $C$ to first equation}\\ m(...
0
votes
1answer
25 views

Finding Constants on a Differential Equation

Question (from my sample exercises in calculus): Find constants $A, B$ and $C$ such that the function $$ y=A\sin x+B\cos x $$ satisfies the differential equation $$ y''+y'-2y=\sin x. $$ I am ...
0
votes
1answer
65 views

How to prove the following formula is a constant?

For any $\rho\in\mathbb{R}^+$, prove that the following formula equals a constant: $$\dfrac{1}{\rho^2}{\int_{-\rho}^\rho x^2 e^{\left(\tfrac{\rho^2}{x^2-\rho^2}\right)}dx}\left({\int_{-\rho}^\rho e^{\...
0
votes
0answers
20 views

Universal Parabolic Constant

I recently learned of a constant that arises in parabolas, similar to that of $\pi$ for circles. Like $\pi$ being the ratio of the circumference of the circle to its diameter, this constant $\sqrt2+\...
4
votes
3answers
174 views

Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
2
votes
0answers
49 views

Why only a handful of distinct mathematical constants appears in most applications?

Is there some theoretical reasoning behind the fact that the same numbers ($\pi$, $e$, $\gamma$, etc) appear again and again in every application in every field of Mathematics? Most people just ...
5
votes
3answers
115 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
0
votes
3answers
45 views

Can you have many integrations for a single function? (sorry if the terminology is wrong)

Forgive me for the amateur question (first time here), but it's bugging me! OK, for example, integrating 1/5x would yield what? ln(5x)/5 +C OR ln(x)/5 +C ? Both of these when differentiated form 1/(...
0
votes
2answers
78 views

Find the limit $\lim_{h\to 0}\frac{ \sqrt{7(a+h)}-\sqrt{7a} }{h} $ in terms of $a$ .

Find the limit in terms of the constant $a$ : $$ \lim_{h \to 0}\frac{\sqrt{7(a+h)}-\sqrt{7a}}{h} $$ I have tried to solve this by multiplying the square roots to both sides but i simply can't solve it ...
0
votes
1answer
39 views

Moment generating function of a constant

This might be trivial, but can you elaborate why moment generating function for a constant $c$ is $e^{cX}$, where $X$ is a random variable.
1
vote
1answer
22 views

Solve series with two constants and opposite exponents

How can I find the general formula for the sum of this series? $$ \sum_{i=0}^n a^ib^{n-i} $$ Where $a$ and $b$ are unrelated constants? I don't think you can split it into $ \sum_{i=0}^n a^i $ and ...
5
votes
1answer
86 views

Proving $\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$

How to prove that $$\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$$ Where $F_{n}$ is the Fibonacci Number.
0
votes
1answer
54 views

Sum of infinite series - Catalan constant

Why is this identity true? $$\sum_{k=1}^{\infty} \frac{sin(k\pi/2)}{k^2} = C$$ where $C$ is Catalan constant.
4
votes
1answer
119 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that $$\...
7
votes
1answer
136 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
2
votes
2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for $e$...
0
votes
2answers
65 views

Is that a known “constant”?

I noticed, using a calculator, that the following operation: $$Y = \sin(\cos(\tan(\log(n))))$$ for values of $n$ from $0.00000001$ to $99999999$, I obtain always quite the same number, id est $$Y = ...
1
vote
2answers
41 views

Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?

I believe that the sum $$\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this ...
0
votes
1answer
28 views

Taking derivatives of constants and variables

When implicitly differentiate a function, for example, $f(x)=(G)(x)$, where G is a constant, is it possible to differentiate it such that we can treat G as a variable? From my understanding it is ...
0
votes
0answers
43 views

Show that $\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$

Let $p(n)$ denote the probability that a randomly chosen one-to-one function $f:\lbrace 1,2,3...,n \rbrace \rightarrow \lbrace 1,2,3...,n \rbrace$ has a least one fixed point. (least one integer $k$ ...
2
votes
0answers
56 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
0
votes
0answers
23 views

Finding the constant in a joint density function

I have been having a bit of difficulty with the following example, whenever i integrate the terms vanish. Any help is appreciated! The joint density function of X and Y is given by $$f(x , y) = $$\...
0
votes
1answer
44 views

Find the constants to make it continuous.

I am having trouble starting this question. Find the constants (a, b or c) to make the function continuous. f(x)= { ...
3
votes
1answer
52 views

Polyhedral symmetry in the Riemann sphere

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
1
vote
0answers
20 views

Use of the constant of integration

I have a differential equations question: Given the differential equation $$t(t+1) \frac{ds}{dt} = sln(s)$$ it can be trivially solved to be $$ln ln s = ln(t) - ln(t+1) +C$$ As known, C is the ...
0
votes
1answer
27 views

General question regarding constants and variables

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to relations and functions. To be specific, I'm stuck on ...
0
votes
0answers
30 views

Max n value for $\frac{1}{x^x}=y^y+n$?

I was looking at the graph of $$\frac{1}{x^x}=y^y+n$$ and found that it disappears at n greater than about $0.752$. What is the exact constant definition for n and why does the graph disappear (...
1
vote
2answers
64 views

What's the value of $\sum_{k=1}^{\infty}(\zeta(2k+1)-\zeta(2k+2))$?

I'm confused about this. I have this expression $$ \frac{1}{2}=\sum_{k=1}^{\infty}(\zeta(2k)-\zeta(2k+1)) $$ Now if I want claculate $\zeta(2)$ I'll do the apropriate manipulations to get $$ \zeta(2)=\...
4
votes
0answers
85 views

Find a constant $C_p$ that satisfies $|f(x+p)-f(y+p)|\le C_p|f(x)-f(y)|$

Let $B^n$ be the unit open ball in $\mathbb{R}^n$, $p\in \mathbb{R}^n$ and $f\colon \mathbb{R}^n\to B^n$ defined as $f(x)=\frac{1}{1+|x|}x$. I believe there are constants $C_p>0$ such that $|f(x+p)...
3
votes
1answer
116 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: $$\begin{align}{_2F_1}\...
5
votes
3answers
80 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
3answers
392 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 ...
2
votes
1answer
69 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
133 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
95 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.