## Encyclopedia > Euler-Mascheroni constant

Article Content

# 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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Grateful Dead ... well. The 1970s live album Live Dead did capture more of their essence, but commercial success did not come until the country influence came through, on American Beauty and ...