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 [math]\displaystyle{ \gamma }[/math] (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 [math]\displaystyle{ \gamma }[/math] is about [math]\displaystyle{ 0.5772156649 }[/math].[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]
- [math]\displaystyle{ \gamma = \lim_{t \to \infty} \left(\sum_{n=1}^{t} \frac{1}{n} - \log(t)\right) }[/math]
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]
- [math]\displaystyle{ \gamma = \int_{1}^{\infty} \left(\frac{1}{\lfloor t \rfloor} - \frac{1}{t}\right) \mathrm{d}t }[/math]
The gamma constant is closely linked to the Gamma function,[6] specifically its logarithmic derivative, the digamma function, which is defined as
- [math]\displaystyle{ \mathrm{\Psi}_0(x) = \frac{\mathrm{d}}{\mathrm{d}x} \log(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)} }[/math]
For [math]\displaystyle{ x=1 }[/math], this gives[6]
- [math]\displaystyle{ \mathrm{\Psi}_0(1) = -\gamma }[/math]
Using properties of the digamma function, [math]\displaystyle{ \gamma }[/math] can also be written as a limit.
- [math]\displaystyle{ -\gamma = \lim_{t \to 0} \left(\mathrm{\Psi}_0(t) + \frac{1}{t}\right) }[/math]
Euler–Mascheroni Constant Media
abm(x) = γ−x
References
- ↑ Euler, Leonhard (1735). De Progressionibus harmonicus observationes (PDF). pp. 150–161.
- ↑ 2.0 2.1 "Greek/Hebrew/Latin-based Symbols in Mathematics". Math Vault. 2020-03-20. Retrieved 2020-10-05.
- ↑ 3.0 3.1 Weisstein, Eric W. "Euler-Mascheroni Constant". mathworld.wolfram.com. Retrieved 2020-10-05.
- ↑ 4.0 4.1 "Euler-Mascheroni Constant | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-05.
- ↑ Sandifer, Edward (October 2007). "How Euler Did It - Gamma the constant" (PDF). Retrieved 26 June 2017.
- ↑ 6.0 6.1 6.2 "The Euler Constant" (PDF). April 14, 2004. Retrieved June 26, 2017.