Euler's constant
| Binary | 0.1001001111000100011001111110001101111101… |
| Decimal | 0.5772156649015328606065120900824024310421… |
| Hexadecimal | 0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A… |
| Continued fraction |
[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ] (It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic. Shown in linear notation) |
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ).
It is defined as the limiting difference between the harmonic series and the natural logarithm:
Here, ⌊x⌋ represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is
The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.
The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):
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