Questions tagged [eulers-constant]

For questions related to Euler's constant $\gamma$, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

Filter by
Sorted by
Tagged with
2
votes
1answer
69 views

How to visualize Euler's number?

I am interested if there is geometric meaning (using graphs) of $(1 + \frac{1}{n})^n$ when $n \rightarrow \infty$. Also, is there visual explanation of why is $e^x = (1 + \frac{x}{n})^n$ when $n \...
2
votes
1answer
36 views

Imaginary numbers calculation for DFT

I am trying to understand the Fourier transformation and the math behind it, so I was trying to use this formula: $$ x_k = \sum_{n=0}^{N-1} x_n e^{-\frac{j2{\pi}kn}{N}} $$ to calculate all $ x_{0\...
1
vote
0answers
56 views

Proving $\int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1-xy)\ln(xy)} \mathrm dy \mathrm dx =\gamma$

The integral is $\int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1-xy)\ln(xy)} dydx $ This is the special case for the Hadjicosta's formula for $s\to -1$ . The proof of which was done by Jonathan Sondow. I ...
1
vote
2answers
49 views

How to solve $\sum _{n=0}^{\infty }\:\frac{n^a}{n!}$ for any $a \geq 1$ in general?

While doing a problem I encountered this summation $$\sum _{n=1}^{\infty }\:\frac{n^2}{n!}=2e.$$ I used $$\sum _{n=0}^{\infty }\:\frac{n}{n!}=e$$ and $$\sum _{n=0}^{\infty }\:\frac{(n-1)^2}{n!}=...
1
vote
1answer
51 views

The Cauchy principal value of the Riemann Zeta function

In many online sources, you can find $$ \zeta(s) \overset{C.P.}{=} \lim_{\epsilon \to 0} \left(\frac{\zeta(s+\epsilon)+\zeta(s-\epsilon)}{2}\right). $$ This seems quite logical, but I neither know how ...
3
votes
3answers
73 views

Integral representations of the Euler–Mascheroni constant

I am trying to verify $$\int_0^\infty te^{-t}\log{t}\,\mathrm dt=1-\gamma,$$ where $\gamma$ is Euler–Mascheroni constant. This problem is exercises 10.2.11(b) in Mathematical methods for physicists, ...
1
vote
1answer
49 views

Prove that $\gamma^\Omega$ is larger than $\Omega^\gamma$, where $\gamma$ is the Euler-Mascheroni constant and $\Omega$ is the omega constant

In this post we denote the Euler-Mascheroni constant as $\gamma$ and the known as omega constant as $\Omega=W(1)$ with $W(x)$ the main/principal branch of the Lambert $W$ function. I don't know if the ...
1
vote
1answer
30 views

A class of generalized Integrals involving polygamma functions

I recently came across some nice integrals and I have a few questions about them: You may have heard of the Euler-Mascheroni constant $\gamma$ and if you did so, you may know the following integral: ...
0
votes
0answers
28 views

Using pentration with non-integer values: how to solve 2^^^1.5? [duplicate]

Say we define using hyperoperators with non-integers the same way we do with exponentiation, that is to say we convert the 'exponent' to a fraction and raise the base to the hyper-power of the ...
0
votes
1answer
31 views

A question about the constant in $\sum_{2\leqslant n\leqslant x}\frac{1}{n\log n}=\log(\log x)+B+O\left(\frac{1}{x\log x}\right)$

I could prove the major part of this identity as the following: $$ \begin{align*} \sum_{2\leqslant n\leqslant x}\frac{1}{n\log n} &=\frac{1}{2\log 2}+\int_2^x \frac{1}{t\log t}dt-\int_{2}^...
0
votes
0answers
12 views

Can a left-endpoint Riemann sum be described by a step function? How would you integrate it?

Can the left endpoint Riemann sum be described by a decreasing step function? I want to find the purple area in the graph from scratch using integration. How would I do it?
2
votes
0answers
41 views

Relationships between irrational numbers

Euler’s formula relates $e$ and $\pi$ (also $0$, $1$ and $i$). Are there relationships between other irrational numbers? Indeed, are there families of related irrational numbers?
1
vote
0answers
32 views

Evaluating a limit to get an integral for Euler's Constant

I have been reading a book on Number Theory and just came across an equation that I can't seem to see how to get the result, in the book is states that: $$\lim_{x \to \infty}\left ( \sum_{n\leq x} \...
0
votes
2answers
42 views

Is the number $1.201~943\dots$ of any significance?

I ran the function $(x^2+y^2)^z=z$, where $z$ is a constant. The function produced a circle, and the $z$ value where the radius of the circle turned out to be the largest was $e$. At that point, the ...
1
vote
3answers
50 views

How do I prove that this limit is equal to e without L'Hospital?

I'm solving some limits and in one of my examples I need to use the fact that: I am, however, unable to prove that this is actually true. I believe I can't just substitue t=(1/x) because than we can ...
11
votes
1answer
238 views

Closed form of Euler-type sum over zeta functions $\sum _{k=2}^{\infty } \frac{\zeta (k)}{k^2}$?

Revisiting the question on the integral over the harmonic number I stumbled over the nice formula $$\sum_{k\ge2} (-1)^{k+1}\frac{\zeta(k)}{k} = \gamma\tag{1}$$ where $\zeta(z)$ is the Riemann zeta ...
3
votes
2answers
791 views

An approximation related to Euler's constant and the Harmonic number

Let's consider Euler's constant $\gamma$, i.e., $$\gamma=\lim_{n\to \infty} \sum_{k=1}^n\frac{1}{k}-\ln(n).$$ Prove the following approximation: $$\sum_{k=1}^{m-1}\frac{1}{k}-\ln(m)+\frac{1}{...
1
vote
0answers
21 views

Distributional integral involving $\frac{e^{i\omega x}}{\omega}$

I saw in a paper the following integral \begin{align} \int_0^\infty \frac{e^{i\omega x}}{\omega}\,d\omega = -\gamma+\frac{i\pi}{2}-\ln(x+i0)\,, \end{align} where $\gamma$ is the Euler's constant and $...
1
vote
3answers
78 views

How to show that the integral $\int_0^1 \ln(1-x^{1/n})dx=-\sum_{k=1}^n\frac{1}{k}$

The integral is $$\int_0^1 \ln(1-x^{1/n})dx=-\sum_{k=1}^n\frac{1}{k}$$ I tried integral by part, but cannot get the $\frac{1}{k}$ term. How can we do this integral? Any hint would be appreciated.
0
votes
0answers
28 views

Calculating large exponential shares / probabilities

Let there be an event space ES. Let there be some sets of objects OS[]. The probabilities of selecting any object are mutually disjoint. Now, assume that the size of each set is based on a number X[...
0
votes
0answers
63 views

Why does Euler's number occur in the Chudnovsky Formula for calculating $\pi$?

The Chudnovsky Formula for calculating $\pi$ reads: $$\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)!(k!)^3 640320^{3k + 3/2}}$$ If you double the second ...
2
votes
2answers
75 views

Proving the convergence of an infinite product

I’m trying to prove the Taylor series of $e$ using binomial expansion: $$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots$$ The steps I’ve tried so far are: $$\left(1+\frac{1}{n}\right)^...
1
vote
0answers
56 views

Proving that the constant in Mertens' third theorem is $e^{-\gamma}$

Suppose I already know that Mertens' third theorem holds in the form $$\prod_{p\leq x} \left(1 - \frac{1}{p}\right) = \frac{C}{\ln x} + O\left(\frac{1}{\ln^2 x}\right)$$ for some constant $C$. I ...
0
votes
2answers
35 views

Derivative power rule conditions

I'm kinda a newbie in calculus, but what are the conditions for the power rule to happen? For example, if we have the number $e$, with the property $\frac{d}{dt}{ {e^t} } = {e^t}$ we can get, using ...
0
votes
1answer
53 views

Chebyshev's Inequality and Euler's constant question clarification

Here's a question to which I don't understand the answer to: In particular: 1) I don't understand why do they choose epsilon to be between 0 and 1? Is there a specific reason i'm missing? 2)Why ...
2
votes
3answers
37 views

Why is this proof of the irrationality of $\gamma$ wrong?

By definition, $$\gamma=\lim_{n\to \infty}\left(\sum_{k=1}^n \left(\frac{1}{k}\right)-ln(n)\right)$$ It is clear that $\sum_{k=1}^n \left(\frac{1}{k}\right)$ will always be rational, as it is a sum ...
0
votes
4answers
76 views

Let $I_n = \int_0^1 x^ne^{-x} dx$, show that $0 < I_n < \frac{1}{n+1}$

The question ask us to use the fact that if $f(x) < g(x)$, $\forall x \in [a,b]$, then $\int_a^b f(x) dx < \int_a^b g(x) dx $. I have no idea about how to choose $f(x)$ so that $\int_0^1 f(x) ...
1
vote
1answer
56 views

Euler-Mascheroni constant trivial [duplicate]

Demonstrate the following trivial equation: $$\int_0^{\infty} \int_0^{\infty}\frac{\log x \log y}{\sqrt{xy}} \cos(x + y)dxdy = (\gamma + 2 \log 2) \pi^2$$ where $\gamma$ is the Euler-...
0
votes
1answer
26 views

complex Fourier series expansion in terms of sinc function

Fourier Expansio over Periodic Function How can I expand $\exp(ikx)$ over $-N/2$ to $N/2$? I have some idea that Euler Formula and geometric series is used but i stuck to find exactly the same thing ...
1
vote
2answers
54 views

Cauchy problem: $x'=\frac{x}{t^2+1}, x(0)=1$.

$$x'=\frac{x}{1+t^2}, \qquad x(0)=1$$ I know the solution to the problem, but I don't get to the right solution myself. My solution: $$\int \frac{dx}{x}=\int \frac{dt}{1+t^2}$$ $$\ln(x)=\cot(t)+ C$...
0
votes
0answers
71 views

Why when we try to solve $f = f '$, do we set the $f(0) = 1$ condition? This is regarding finding Euler's number.

I was trying to get a better understanding for e and pi, and came across Alon Amit's explanation here: https://www.quora.com/q/bzxvjykyriufyfio/What-is-math-pi-math-and-while-were-at-it-whats-math-e-...
3
votes
1answer
121 views

A scary looking relationship between Euler constant $\gamma$ and $\pi$

I saw this scary looking relation between the Euler constant $\gamma$, the Gamma function and $\pi$ in Peter Luschny blog. Any proofs? $$ \left({{\frac {\Gamma (\gamma )}{\Gamma (2\gamma )\Gamma (1/...
0
votes
1answer
58 views

How to simplify the expression $\frac{\ln(t)}{\ln(t')}?$

I am solving a Cauchy problem: $x'+\frac{x}{t\ln t}=\frac{1}{\ln t'}, \ \ x(e)=e$ and I came to an expression $\frac{\ln t}{\ln t'},$ that I don't know how to simplify or express differently. Can ...
0
votes
1answer
32 views

Is there any significance for $\sqrt[e\cdot\log\left(x\right)]{x}$?

I get that; $$\sqrt[e\cdot\log\left(x\right)]{x} = 2.332810391$$ For $x ≠ 1,0,-1,-2,...$. Why is this and does this hold any significance?
0
votes
1answer
31 views

What's the links between factorial,Basel Problem and Euler constant .

I'm studying the factorial function and I have found the following identity : $$\frac{\partial^2 x!}{\partial x^2}=-2\gamma+\gamma^2+\frac{\pi^2}{6}$$ At $x=1$ But the Basel problem says : $$\...
0
votes
2answers
73 views

What is the theorem that approximate a “success” 1 out N probability tried N times, 2N times, etc, for not succeeding at least once?

That is, if the probability of success is 1/450, so failing means 449/450, then if we try it 450 times, then the chance of failing (not succeeding even one time) is: $$ (449/450) ^ {450} = 0....
0
votes
1answer
109 views

Fake proof that $\zeta (1)=\gamma$

The following proof is fake, but I have no idea why it's not right (I know it's paradoxical, since $\zeta (1)$ diverges): $$\begin{align*}\zeta (1)&=\dfrac{2\zeta (1)}{2}\\&=\dfrac{\...
6
votes
2answers
187 views

Prime Zeta function at 1

I wanted to find $$\lim_{s\to 1} (P(s)-\ln(\zeta(s)))$$ and here is my attempt: So we know that $$M=\gamma +\sum_{n=2}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}$$ and that $$P(s)=\sum_{n=1}^\infty \mu(...
5
votes
5answers
318 views

How can I prove $\frac{\gamma}{2}=\int_{0}^{\infty}\frac{e^{-x^{2}}-e^{-x}}{x}\text{d}x$?

How can I prove the following equation? $$\frac{\gamma}{2}=\int_{0}^{\infty}\frac{e^{-x^{2}}-e^{-x}}{x}\text{d}x$$
3
votes
0answers
102 views

Prove a new digamma finite sum

There are a few interesting finite sums of digamma of a rational argument listed on Wikipedia (from this paper). One of them is the following: $$\sum _{m=1}^{N-1}\psi \left({\frac {m}{N}}\right)\cdot ...
2
votes
2answers
75 views

Integral involving $\ln$ and $\gamma$

I want to know if its possible to have a closed form of this integral $$\int_0^\infty e^{-x}\ln(kx) dx$$ I know that if k = 1 then the integral is equal to $-\gamma$ but i want to find a generalized ...
0
votes
0answers
65 views

$\pi~$ expanded in terms of the Euler-Mascheroni Constant $\gamma$

Is there a known expansion of $\pi$ as a function of the Euler-Mascheroni $\gamma$? As in, $$f(\gamma)=\pi h(\pi,\gamma)$$ where $\gamma$ appears alone with only rational arguments like $f(1/2,\gamma)$...
4
votes
3answers
83 views

Integral representation of the Euler-Mascheroni constant involving $\pi$

A month ago, I came up with a proof that $\gamma = \frac12 + \int_0^{\frac1\pi} \arctan(\cot(\frac1x)) \,dx$ where $\gamma$ is the Euler-Mascheroni constant and $\arctan$ is the inverse $\tan$ ...
0
votes
1answer
37 views

how to calculate this euler equation or e of P(X)?

I cannot understand how the below equation produce this result? Can anyone please explain the steps for this calculation? I tried finding the exponential values but it gives 2.36*10-06 What is our ...
0
votes
1answer
58 views

Convergence of the sequence $\left\{x_n\right\}$ where $x_n = \frac{1}{1.3}+\frac{1}{2.5}+…\frac{1}{n(2n+1)}$ [duplicate]

Let $\left\{x_n\right\}$ be a sequence where $x_n = \frac{1}{1\cdot3}+\frac{1}{2\cdot5}+...\frac{1}{n\cdot(2n+1)}$ I have to calculate, to which point does the sequence $\left\{x_n\right\}$ converge, ...
1
vote
1answer
82 views

Is it possible to compute this using Euler? $\frac{(\cos 1 + \cos 89)(\cos 2+\cos 88)\cdots(\cos 44 + \cos 46)}{\cos 1\cos 2\cdots\cos 44}$

Is it possible to compute this using Euler? $$\dfrac{(\cos(1) + \cos (89))(\cos(2)+\cos(88))...(\cos(44) + \cos(46))}{\cos(1)\cos(2)...\cos(44)}$$ I have easily computed this problem using ...
0
votes
1answer
73 views

How can I prove this limit converges to the Euler-Mascheroni constant?

I'm trying to prove that $$\lim_{N\to\infty}\;2\left[ \int_0^N \frac{\text{erf}(x)}{x}\,dx - \ln(2N) \right] = \gamma,$$ where $\gamma$ is the Euler-Mascheroni constant. This looks similar to the ...
0
votes
2answers
104 views

About the Euler-Mascheroni Constant

Okay, so the limiting difference between the harmonic series and the natural logarithm is known as the Euler-Mascheroni constant, $\gamma= 0,577$. My question is: is there any base for the logarithm ...
-1
votes
1answer
70 views

Is $e^{ix}$ just the name of a point on the unit circle?

Am I right to say that $e^{ix}$, where $x$ is the angle in a unit circle, is just the name of a point on the unit circle corresponding with some angle?
1
vote
1answer
58 views

Continued fraction $[0;2,6,10,14,…,2(2n-1)] = \frac{e-1}{e+1}$

I would like to ask about the following relation, I wonder how to reach it. \begin{equation} K_{n=1}^{\infty} \frac{1}{2(2n-1)} = \frac{1}{2+\frac{1}{6+\frac{1}{10+\cdots}}} = \frac{e-1}{e+1} \...

1
2 3 4 5
7