Euler–Mascheroni constant
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, [math]\displaystyle{ \gamma }[/math], 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 [math]\displaystyle{ \gamma }[/math] is about [math]\displaystyle{ 0.5772156649 }[/math]. 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]:
- [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, which gives the greatest integer less than or equal to a given number.
- [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 [3], 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 us[3]
- [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]