Euler–Mascheroni constant

In mathematics, Euler-Mascheroni constant is a number that appears in analysis and number theory. It first appeared in the work of Swiss mathematician Leonhard Euler in the early 18th century.^[1] It is usually represented with the Greek letter ${\displaystyle \gamma }$ (gamma),^[2] although Euler used the letters C and O instead.

It is not known yet whether the number is irrational (that is, cannot be written as a fraction with an integer numerator and denominator) or transcendental (that is, cannot be the solution of a polynomial with integer coefficients).^[3] The numerical value of ${\displaystyle \gamma }$ is about ${\displaystyle 0.5772156649}$.^[4][3] Italian mathematician Lorenzo Mascheroni also worked with the number, and tried unsuccessfully to approximate the number to 32 decimal places, making mistakes on five digits.^[5]

It is significant because it links the divergent harmonic series with the natural logarithm. It is given by the limiting difference between the natural logarithm and the harmonic series:^[2][6]

${\displaystyle \gamma =\lim _{t\to \infty }\left(\sum _{n=1}^{t}{\frac {1}{n}}-\log(t)\right)}$

It can also be written as an improper integral involving the floor function, a function which outputs the greatest integer less than or equal to a given number.^[4]

${\displaystyle \gamma =\int _{1}^{\infty }\left({\frac {1}{\lfloor t\rfloor }}-{\frac {1}{t}}\right)\mathrm {d} t}$

The gamma constant is closely linked to the Gamma function,^[6] specifically its logarithmic derivative, the digamma function, which is defined as

${\displaystyle \mathrm {\Psi } _{0}(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\log(\Gamma (x))={\frac {\Gamma '(x)}{\Gamma (x)}}}$

For ${\displaystyle x=1}$, this gives^[6]

${\displaystyle \mathrm {\Psi } _{0}(1)=-\gamma }$

Using properties of the digamma function, ${\displaystyle \gamma }$ can also be written as a limit.

${\displaystyle -\gamma =\lim _{t\to 0}\left(\mathrm {\Psi } _{0}(t)+{\frac {1}{t}}\right)}$