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

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-2
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0answers
36 views

Theoretical Limit to the Physical Measurement of Pi [on hold]

A. Is there a theoretical limit to the accuracy of the pyhsical measurement of Pi in a physical circle that is based upon the Plank length? B. Would such a limit, if it existed as posited in A, ...
1
vote
0answers
23 views

Quartic Polynomial Manipulation

I have a quartic polynomial in $x$ (too long to write here) $f(x,c_1, c_2, c_2)$ where $c_1, c_2, c_3$ are constants which I have complete freedom over how to fix their values, as long as they are ...
0
votes
1answer
12 views

Landau Constants precision

Looking at an article in Wikipedia on Landau constants it indicates that the actual values are not known except that they are within a certain interval. This seems surprising to me since for most ...
1
vote
1answer
60 views

How is PI used to predict weather patterns?

I've heard that using PI to predict weather patterns is possible. I would like verification on this, and how this is possible. I can't seem to find any other sources explaining this concept. My ...
0
votes
0answers
25 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 ...
1
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1answer
54 views

Why we can't define more mathematical constant?

I would like to know how many mathematical constant are there? I saw this link and I know the names. Who can define a mathematical constant? Someone can say that ...
0
votes
0answers
37 views

Basic doubt about derivative

given the following equation: $f(x) = (x-1)\arctan x$, I am to calculate its Taylor polynomial of 2nd degree with remainder of Peano. I began by calculating the derivative, using the product rule: ...
3
votes
1answer
34 views

A misconception about arbitary constant

Given a function $f(x)$ from $\mathbb R$ to $\mathbb R$, If $f'(x)=0$ $\text{ for all } x\in \mathbb R$. Then $f(x)=C$.(This is my understanding) Question: I think that $C$ has to remain constant ...
8
votes
0answers
238 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
2
votes
4answers
174 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
0
votes
2answers
51 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
1
vote
1answer
50 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
3answers
37 views

How to prove that $\frac{d}{dx}\epsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant

How to prove that $\frac{d}{dx}\varepsilon x^2=2\varepsilon x$ where $\varepsilon$ is just a constant? Thanks in advance for your immense help.
0
votes
1answer
37 views

What is the constant $c$ in $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy}$?

In the manner in which $\frac{d}{dx} e^{x} = e^{x}$. What is the value of the constant $c$ for which $\frac{d}{dx} \frac{d}{dy} c^{xy} = c^{xy} $ ?
1
vote
1answer
50 views

Asymptotic Big Omega Proof

Let $f(n) = 2n^4 − 4n^3 + 16n^2 − 64^n + 3$. (a) Using the constant $c = 1.9$, prove that $f(n) \in \Omega(n^4)$. Just by looking at it it is clear to me that this is true as for $f(n) \in ...
1
vote
4answers
170 views

Calculus Derivative - Finding unknown constants

Determine the constants $a$, $b$, $c$, and $d$ so that the curve defined by $y = ax^3 + bx^2 + cx + d$ has a local maximum at the point $(2, 4)$ and a point of inflection at the origin. Sketch the ...
0
votes
2answers
34 views

Statistics - scaling sets of data - mean & S.d

A set of numbers has a mean of 22 and a standard deviation of 6. If 3 is added to each number of the set, and each resulting number is then doubled, find the mean(50) and standard deviation(12) of the ...
5
votes
2answers
100 views

If $\pi $ is a normal number, is $\tau $ one?

If $\pi$ is a normal number, would that imply that $\tau =2\pi $ is also a normal number? If so, why? Something tells me that it should be, but I have no idea how to prove it. If all digits of $\pi$ ...
1
vote
5answers
70 views

Limits to infinity Finding Constant Number

Hi I have a question regarding of limits to infinity please help which I need to find the constant number for a and b. Please help! Thank You! The question states the user to find the following ...
0
votes
3answers
236 views

What are the uses of Euler's number $e$?

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ...
0
votes
1answer
40 views

Integration involving multiple constants…

So I've been tackling the rather nasty integral of... $\int^R_s\frac{2r}{\sqrt{r^2-s^2}}.\frac{1}{2}(R-r)^2dr$ ...where R and s are constants. However, every method I try I seem to get stumped by ...
6
votes
2answers
101 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can proof that the two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and ${d\over dx}\ln x$, some uses the former ...
1
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0answers
23 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
0
votes
0answers
33 views

$f$ is analytic with range as a circle

I was given that range of $f$ lies on a cirlce, and $f$ is analytic on $D$. I want to show that $f$ is constant. This is my approach: I suppose that $f$ lies on a circle $|w-P|=R$, where $P,R$ are ...
5
votes
1answer
100 views

On a constant defined by Ramanujan.

In the second letter to Hardy Ramanujan writes about the number of prime numbers less than $n$ there he writes. Here this constant $\mu$ facinated me . What is its closed form? and How to compute ...
3
votes
0answers
76 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
84 views

Express the mathematical constant $e$ in terms of a limit that goes to zero.

The mathematical expression of the mathematical constant $e$ in terms of a limit that goes to infinity is $$e = \lim\limits_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$ But can we express the ...
8
votes
3answers
416 views

On the “Look-and-Say” sequence and Conway's constant

The look-and-say sequence starting with $S_1=1$ is, $$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$ If $L_n$ is the number of digits of the $n$th term then, ...
-1
votes
1answer
139 views

Why is the number Pi more popular than any other constant? [closed]

What is so special about the number $\pi$? There are many more interesting constants, such as e, $\gamma, \sqrt{2}$ or the catalanian number. $\pi$ has been calculated to more digits than any other ...
1
vote
3answers
76 views

General form for the series expansion of $e$

I've found a lot of series expansions of the Napier's constant. I was wondering if a general form for this could be devised. They all have a trend and similarities. I've been trying but I've been ...
3
votes
1answer
123 views

Why $\operatorname e$ was named e ? What is the history and reason behind it?

Why the constant that Euler discovered has symbol $\operatorname e$ and not other symbols? What is the history and reason behind it? Thanks.
8
votes
4answers
354 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
0
votes
1answer
54 views

Prove that the function is constant $f(x)=\arcsin2x\sqrt{1-x^{2}} - 2\arcsin x$ [closed]

Prove that this function is constant: $$f(x)=\arcsin\left(2x\sqrt{1-x^{2}}\right) - 2\arcsin x$$
2
votes
1answer
43 views

Antiderivative where resulting constant depends on x?

Everything was going really well until one week before the exam when the teacher gave us this problem: $\int \frac{2x^2+13x+19}{x^2+5x+6} dx$ For which I and Wolfram Alpha finds the solution: ...
22
votes
2answers
280 views

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as ...
2
votes
0answers
75 views

In the Hunt for Kaprekar's Constants for more than 4 digits.

Kaprekar's constant is $6174$ . Take any four digit number with at least two different digits; create two four digit numbers by writing the digits in descending order and in ascending order; subtract ...
1
vote
1answer
47 views

Mathematical constants

Am I mistaken, or is there a mistake on the mathematical constants Wikipedia page that describes the Conic constant / Schwarzschild constant in terms of Napier's constant?
0
votes
1answer
56 views

Real Analysis constant function

Suppose that $f:(a,b) \to \Bbb R$ satisfies $|f(x)-f(y)|=M|x-y|^\alpha$ for some $\alpha > 1$, some $M \geq 0$, an all $(x,y)$ in $(a,b)$. Prove $f$ is constant on $(a,b)$ I have absolutely no ...
3
votes
0answers
310 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 who's author has passed away. ...
0
votes
1answer
62 views

What denotes the essence of a function object in mathematics?

In other words, when does something become a function, and why? Take this, for example: x = y (x + z) = 350 Is anything enclosed within the brackets considered ...
2
votes
1answer
46 views

Do I need different constant names for different levels of integration?

I'm just getting my feet wet in integration, so pardon me if I misuse a term. Let's take the anti-triple-derivative (I'm not sure if that's what it's actually called) of $8x$. $y'''=8x$ ...
-4
votes
1answer
44 views

What is the Constant

If a car is travelling at 55 mph. What is the algebraic expression showing how far the car travelled for a number of hours. What is the constant?
2
votes
1answer
75 views

Can the Landau-Ramanujan Constant be generalized towards cubes?

The Landau-Ramanujan Constant is related to the sum of 2 squares. See : http://en.wikipedia.org/wiki/Landau%E2%80%93Ramanujan_constant Can a similar thing be said for the sum of 4 positive cubes ? Or ...
0
votes
1answer
79 views

Closed Forms of Certain Zeta constants?

The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $Re(s)>1$. I am interested in some particular "irrational " Values of it. Like $\zeta(\pi)=1.176241738...$ ...
1
vote
1answer
55 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?
10
votes
1answer
231 views

Is there an efficient method for the calculation of $e^{1/e}$?

(I wonder whether this is appropriate for the Math StackExchange or whether it'd be better on Stack Overflow as it deals with computing, but I'm asking about mathematical details, not about ...
17
votes
2answers
754 views

Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
1
vote
1answer
89 views

Proving this identity $\gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\zeta(k+1)-1}{k+1}$ where $\gamma$ is the Euler-Masceroni constant

I've seen this identity here $$ \displaystyle \gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\dfrac{\zeta(k+1)-1}{k+1} $$ and I'd like to know how it is deduced. Could anyone help? Thanks. ...
2
votes
0answers
44 views

Is $e$ uniformly distributed in all bases?

There has been talk of whether or not $\pi$ is normal, i.e. uniformly distributed in all bases $b$ where $b\ge2$. The general response has been that we expect that it is, and have found no obvious ...
12
votes
4answers
275 views

Are there two $\pi$s?

The mathematical constant $\pi$ occurs in the formula for the area of a circle, $A=\pi r^2$, and in the formula for the circumference of a circle, $C= 2\pi r$. How does one prove that these constants ...