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Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the Euler–Mascheroni constant), and the standard deviation is /
where is the Euler–Mascheroni constant. The sum converges for all complex z {\displaystyle z} , and we take the usual value of the complex logarithm having a branch cut along the negative real axis.
where is the Euler–Mascheroni constant, exp(x) = e x is the exponential function, and Π denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is
The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant. In mathematics , the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function :
The interpolating function is in fact closely related to the digamma function = (+) +, where ψ(x) is the digamma function, and γ is the Euler–Mascheroni constant. The integration process may be repeated to obtain H x , 2 = ∑ k = 1 ∞ ( − 1 ) k − 1 k ( x k ) H k . {\displaystyle H_{x,2}=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{x ...
As well as the two cases used by Beukers to get alternate expressions for ζ(2) and ζ(3), the formula can be used to express the Euler–Mascheroni constant as a double integral by letting s tend to −1: = ( ()).
where γ is the Euler–Mascheroni constant. Actually, ψ is the only solution of the functional equation (+) = + that is monotonic on R + and satisfies F(1) = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence equation and convexity restriction. This implies the useful difference equation: