Euler–Mascheroni constant

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The 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, , although Euler used the letters C and O instead. It is not known yet whether the number is irrational (which would mean that it cannot be written as a fraction with an integer numerator and denominator) and/or transcendental (which would mean that it is not the solution of a polynomial with integer coefficients). The numerical value of is about . 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.[2]

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 [3]:

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

The gamma constant is closely linked to the Gamma function [3], specifically its logarithmic derivative, the digamma function, which is defined as

For , this gives us[3]

Using properties of the digamma function, can also be written as a limit.

References[change | change source]

  1. Euler, Leonhard (1735). De Progressionibus harmonicus observationes (PDF). pp. 150–161.
  2. Sandifer, Edward (October 2007). "How Euler Did It - Gamma the constant" (PDF). Retrieved 26 June 2017.
  3. 3.0 3.1 3.2 "The Euler Constant" (PDF). April 14, 2004. Retrieved June 26, 2017.