# 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
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### 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 ...
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### 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 ...
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### . 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?
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### 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?
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### 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 ...
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### 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 $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, ...
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### 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 ...
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### 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 ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{\,x}\bigg\}\bigg\{\frac{1}{x\,y}\bigg\}dx\,dy\,$$
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### 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 ...