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# Euler-Mascheroni constant

The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm:

$\gamma = \lim_{n \rightarrow \infty } \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx$

Intriguingly, the constant is also given by the integral:

$\gamma = - \int_0^\infty { \ln(x) \over e^x }\,dx.$

Its value is approximately

γ ≈ 0.577215664901532860606512090082402431042159335 9399235988057672348848677267776646709369470632917467495...

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, its denominator has more than 10,000 digits.

The Euler-Mascheroni constant appears, among other places, in:

It is named for the mathematicians Leonhard Euler and Lorenzo Mascheroni.