Euler's constant

Euler's constant

Rationality	Unknown
Symbol	γ
Representations
Decimal	0.57721...
Continued fraction	${\displaystyle \gamma =0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{4+\dots }}}}}}}}}}}}}}}$

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:

$${\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)\\&=\int _{1}^{\infty }\left({\frac {1}{\lfloor x\rfloor }}-{\frac {1}{x}}\right)\,\mathrm {d} x.\end{aligned}}}$$

Here, ⌊·⌋ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is: