Euler-Mascheroni constant
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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:
- <math>\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<math>
History
The constant was first defined by Swiss mathematician Leonhard Euler in a paper De Progressionibus harmonicus observationes published in 1735. Euler used the notation C for the constant, and initially calculated its value to 6 decimal places. In 1761 he extended this calculation, publishing a value to 16 decimal places. In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation γ for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made an error in the 20th decimal place.
Properties
Intriguingly, the constant is also given by the integral:
- <math>\gamma = - \int_0^\infty { \ln(x) \over e^x }\,dx. <math>
It can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:
- <math>\gamma = \sum_{n=2}^{\infty} \frac{(-1)^n\zeta(n)}{n}. <math>
γ can also be calculated as a derivative of Euler's Gamma function:
- <math>\gamma = -\Gamma'(1).<math>
Its value is approximately
- γ ≈ 0.577215664901532860606512090082402431042159335 9399235988057672348848677267776646709369470632917467495...
The constant eγ is also important in number theory,
- <math>e^\gamma <math> = 1.78107241799019798523650410310717954916964521430343...
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.
External links
- Euler-Mascheroni constant at MathWorld (http://mathworld.wolfram.com/Euler-MascheroniConstant.html)
- Euler-Mascheroni Constant from the Mathcad Library (http://www.mathcad.com/library/Constants/euler.htm)ca:Constant d'Euler-Mascheroni
de:Euler-Mascheroni-Konstante es:Constante de Euler-Mascheroni fr:Constante d'Euler-Mascheroni ko:오일러 상수 it:Costante di Eulero - Mascheroni nl:Constante van Euler-Mascheroni ja:オイラーの定数 pl:Stała Eulera sl:Euler-Mascheronijeva konstanta