# 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.

271 questions
29 views

48 views

### What is $\sinh(x)$? And $i\pi$

I've been looking around the internet for a way to solve $\sin(i)$, and I found something about $\sinh(x)$ I furthered my search and found something about $\sinh(x)= \frac{e^x - e^{-x}}{2}$. When I ...
53 views

149 views

20 views

### Integral of the difference between a function and its floor

I know the following identity: $$\int_1^\infty\left(-\frac{1}{x}+\frac{1}{\lfloor x\rfloor}\right)dx=\gamma$$ I have a function $f(b)=0.5\left(\sqrt{(N-b^2)}-b+1\right)$. Can a similar identity be ...
41 views

### . Suppose gcd(a, m) = d and that m > 1. Consider the congruence ax ≡ b (mod m). Should there be a solution for every choice of b?

If yes, prove your claim; if not, give a counter example X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
24 views

### Limit inside the exp function

I do not understand the following limit-rule: $$\lim_{x\to\infty}\exp(f(x))=\exp(\lim_{x\to\infty}f(x))$$ Why is that true?
49 views

### Unicode notation of Euler Constant [closed]

According to Wikipedia, eulers-constant is usually denoted as $\gamma$. Why does Unicode use 'ℇ' to denote this number? Wikipedia redirects 'ℇ' to eulers-constant, but in that page, there's no ...
44 views

### Clever equalities proven similarly to Euler's Identity

From How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, a very elegant proof of Euler's Identity was given. Namely, observing $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, ...
155 views

### What mathematical consequences might there be if Euler Mascheroni constant is rational?

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant. Since we cannot prove ...
85 views

287 views

### Evaluation of $\int_{0}^{1}\int_{0}^{1}\{\frac{1}{\,x}\}\{\frac{1}{x\,y}\}dx\,dy\,$

Let $\{\}$ denote the fractional part function, does the following double integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{\,x}\bigg\}\bigg\{\frac{1}{x\,y}\bigg\}dx\,dy\,$$
58 views

64 views

### What are the roots of the equation $z^{40} - z^{20} - a(a+1)$?
I found this question in a book. The answer given is that the roots are $(a+1)^{1/20}\exp\left({\frac{i2k_1\pi}{20}}\right)$ and $(a+1)^{1/20}\exp\left({\frac{i(2k_1+1)\pi}{20}}\right)$. How do I ...