## Euler's Constant

### Description

Explanation of Euler's Constant.

### Details

Euler's Constant, here denoted ε, is a real-valued number that can be defined in several ways. Johnson et al. (1992, p. 5) use the definition:

ε = \lim_{n \to ∞}[1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n} - log(n)]

and note that it can also be expressed as

ε = -Ψ(1)

where Ψ() 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).

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### References

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.

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