Euler-Mascheroni constant

Named after mathematicians Leonhard Euler (1707—1783) and Lorenzo Mascheroni (1750—1800).

The origin of the notation γ is unclear: it may have been first used by either Euler or Mascheroni.[1] It possibly reflects the constant's connection to the gamma function.

Euler-Mascheroni constant

1. (mathematics) A constant, denoted γ and recurring in analysis and number theory, that is defined as the limiting difference between the harmonic series and the natural logarithm and has the approximate value 0.57721566.
   • 1988, Mathematics Magazine, Volume 61, Mathematical Association of America, page 82,

     The run for ${\displaystyle \gamma }$, the Euler-Mascheroni constant, for instance, yielded 583 approximations with six decimals or more!

   • 2003, Barry Guiduli (translator), János Surányi, Paul Erdős, Topics in the Theory of Numbers, Springer, page 100,

     In the previous section we mentioned that we do no know, for instance, whether the Euler–Mascheroni constant or the numbers ${\displaystyle \zeta _{2k+1}}$ for ${\displaystyle k\geq 2}$ are rational.

   • 2013, Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, Springer, page 252,

     The Euler–Mascheroni constant, ${\displaystyle \gamma }$, considered to be the third important mathematical constant next to ${\displaystyle \pi }$ and ${\displaystyle e}$, has appeared in a variety of mathematical formulae involving series, products and integrals […] .

• Mathematically, ${\displaystyle \textstyle \gamma =\lim _{n\to \infty }\left(-\ln n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx}$, where ${\displaystyle \lfloor x\rfloor }$ represents the floor function.

• (mathematical constant): Euler's constant, gamma