Search results
Results from the WOW.Com Content Network
The summatory of reciprocal totient function is defined as ():= = ()Edmund Landau showed in 1900 that this function has the asymptotic behavior (+ ) + + ()where γ is the Euler–Mascheroni constant,
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 notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to the gamma function. [3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, [ 4 ] and Augustus De Morgan used it in a textbook published in parts ...
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 ...
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 /
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 Euler-Mascheroni constant emerges as the Improper Integral from zero to infinity at the integration on the product of negative Natural Logarithm and the Exponential reciprocal. But it is also the improper integral within the same limits on the Cardinalized Difference of the reciprocal of the Successor Function and the Exponential Reciprocal:
The density of the maximum entropy distribution for this class is constant on each of the intervals [a j-1,a j). The uniform distribution on the finite set { x 1 ,..., x n } (which assigns a probability of 1/ n to each of these values) is the maximum entropy distribution among all discrete distributions supported on this set.