Euler–Mascheroni constant

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(This continued fraction is not 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 γ (gamma).

It is defined as the limiting difference between the harmonic series and the natural logarithm:

Its numerical value to 50 decimal places is

γ should not be confused with the base of the natural logarithm, e, which is sometimes called Euler's number.



The constant first appeared in a 1735 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 because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835.[2]

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