# 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.

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