# Questions tagged [euler-mascheroni-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.

330 questions
Filter by
Sorted by
Tagged with
111 views

### Does this $\int_{0}^{\infty}(\frac{\log x}{e^x})^n dx$ always have a closed form for $n$ being positive integer ? what about its irrationality?

It is known that $\int_{0}^{\infty}\left(\frac{\log x}{e^x}\right)^n dx=-\gamma$ for $n=1$ and for $n=2$ we have :$\frac{1}{12}(\pi^2+6(\gamma+\log 2)^2)$ and for $n=3$ we have this form , What I have ...
67 views

### Why is the irrationality of $\gamma$ so ellusive? (Euler–Mascheroni constant)

Perhaps I'm in way over my head asking this; regardless, considering the abundance of known formulas and properties of $\gamma$, seemingly comparable in quantity to the likes of $\pi$ or $e$, why is ...
79 views

### Prove that $\int_{0}^{\infty} e^{-x}x\log{x}\ dx =1-\gamma$ [closed]

I tried to this equation: $$\int_{0}^{\infty} e^{-x}x\log{x}\ dx =1-\gamma$$ but I was stuck because I couldn't avoid the appearance of $Ei(0)$ and $1/0$ for my bad way. I want to know how to avoid ...
68 views

80 views

73 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 ...
51 views

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?
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?
32 views

21 views

73 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-...
126 views

73 views

201 views

76 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 ...
### $\pi~$ 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 $f(1/2,\gamma)$...
### 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 $\tan$ ...