Explanation of Euler's Constant.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Euler's Constant, here denoted \(\epsilon\), is a real-valued number that can be defined in several ways. Johnson et al. (1992, p. 5) use the definition: $$\epsilon = \lim_{n \to \infty}[1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} - log(n)]$$ and note that it can also be expressed as $$\epsilon = -\Psi(1)$$ where \(\Psi()\) is the digamma function (Johnson et al., 1992, p.8).
The value of Euler's Constant, to 10 decimal places, is 0.5772156649.
The expression for the mean of a
Type I extreme value (Gumbel) distribution involves Euler's
constant; hence Euler's constant is used to compute the method of moments
estimators for this distribution (see eevd).
Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, pp.4-8.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
Extreme Value Distribution, eevd.