Euler-Mascheroni constant

The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm:

Intriguingly, the constant is also given by the integral:

Its value is approximately

γ ≈ 0.577215664901532860606512090082402431042159335 9399235988057672348848677267776646709369470632917467495...

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, its denominator has more than 10,000 digits.

The Euler-Mascheroni constant appears, among other places, in:

• a product formula for the gamma function
• calculations of the digamma function
• calculation of the Meissel-Mertens constant
• expressions involving the exponential integral

It is named for the mathematicians Leonhard Euler and Lorenzo Mascheroni.

External link

• Euler-Mascheroni constant at MathWorld