Euler–Mascheroni constant

In mathematics, 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 (gamma),[2] although Euler used the letters C and O instead.

It is not known yet whether the number is irrational (that is, cannot be written as a fraction with an integer numerator and denominator) or transcendental (that is, cannot be the solution of a polynomial with integer coefficients).[3] The numerical value of is about .[4][3] 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.[5]

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:[2][6]

It can also be written as an improper integral involving the floor function, a function which outputs the greatest integer less than or equal to a given number.[4]

The gamma constant is closely linked to the Gamma function,[6] specifically its logarithmic derivative, the digamma function, which is defined as

For , this gives[6]

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

References

  1. ↑ Euler, Leonhard (1735). De Progressionibus harmonicus observationes (PDF). pp. 150–161.
  2. ↑ 2.0 2.1 "Greek/Hebrew/Latin-based Symbols in Mathematics". Math Vault. 2020-03-20. Retrieved 2020-10-05.
  3. ↑ 3.0 3.1 Weisstein, Eric W. "Euler-Mascheroni Constant". mathworld.wolfram.com. Retrieved 2020-10-05.
  4. ↑ 4.0 4.1 "Euler-Mascheroni Constant | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-05.
  5. ↑ Sandifer, Edward (October 2007). "How Euler Did It - Gamma the constant" (PDF). Retrieved 26 June 2017.
  6. ↑ 6.0 6.1 6.2 "The Euler Constant" (PDF). April 14, 2004. Retrieved June 26, 2017.


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