Euler's constant

Euler's constant
Euler's constant
γ; 0.57721...
General information
Type	Unknown
Fields	Analytic number theory;
History
Discovered	1734
By	Leonhard Euler
First mention	De Progressionibus harmonicis observationes
Named after	Leonhard Euler; Lorenzo Mascheroni;

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 $log$ :

{\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx.\end{aligned}}

Here, $⌊ ⌋$ represents the floor function.

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

0.57721566490153286060651209008240243104215933593992... 

Unsolved problem in mathematics:

Is Euler's constant irrational? If so, is it transcendental?

(more unsolved problems in mathematics)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.