Concept

Euler's constant

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: Here, ⌊ ⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. 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. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842. Euler's constant appears, among other places, in the following (where '' means that this entry contains an explicit equation): Expressions involving the exponential integral The Laplace transform* of the natural logarithm The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants* Calculations of the digamma function A product formula for the gamma function The asymptotic expansion of the gamma function for small arguments. An inequality for Euler's totient function The growth rate of the divisor function In dimensional regularization of Feynman diagrams in quantum field theory The calculation of the Meissel–Mertens constant The third of Mertens' theorems* Solution of the second kind to Bessel's equation In the regularization/renormalization of the harmonic series as a finite value The mean of the Gumbel distribution The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.

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https://en.wikipedia.org/wiki/Euler's_constant

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Official source

https://en.wikipedia.org/wiki/Euler's_constant

About this result

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