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Categories: Mathematical constants | Real numbers

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:

$\gamma = \lim_{n \rightarrow \infty } \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx$

Intriguingly, the constant is also given by the integral:

$\gamma = - \int_0^\infty { \ln(x) \over e^x }\,dx.$

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
the first term of the Taylor series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants.

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

External link

Euler-Mascheroni constant at MathWorld

Categories: Mathematical constants | Real numbers

Science Fair Project Encyclopedia Contents Page

03-10-2013 05:06:04
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