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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:

$\gamma = \lim_{n \rightarrow \infty } \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)$

Intriguingly, the constant is also given by the integral:

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

where ln(x) is the natural logarithm of x.

Its value is approximately

γ ≈ 0.57721566...

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, it has a large denominator.

The Euler-Mascheroni constant appears in

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