Euler-Mascheroni constant

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Euler-Mascheroni constant

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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, János Surányi, Paul Erdős, translated by Barry Guiduli, 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