Search results
Results from the WOW.Com Content Network
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:
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): [1]
The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series .
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
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 definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers: = = (+) /, where is the Euler–Mascheroni constant. [1] This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form.
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 /
and γ ≈ 0.57721, the Euler–Mascheroni constant. The following table gives some overview over the first 2 26 – 1 = 67 108 863 values of the λ function, for both, the exact average and its Erdős-approximation. Additionally given is some overview over the more easily accessible “logarithm over logarithm” values LoL(n) := ln λ(n ...