Definition:Euler-Mascheroni Constant

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Definition

The Euler-Mascheroni Constant $\gamma$ is the real number that is defined as:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \gamma$	$:=$	$\displaystyle \lim_{n \to +\infty} \left({\sum_{k=1}^n \frac 1 k - \int_1^n \frac 1 x \ \mathrm dx}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \lim_{n \to +\infty} \left({H_n - \ln n}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

where $H_n$ is the harmonic series and $\ln$ is the natural logarithm.

The existence of this constant is demonstrated in Existence of Euler-Mascheroni Constant.

Its value is approximately $0.57721\ 56649\ 01532\ 86060\ 6512 \ldots$ This sequence is A001620 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

This constant is otherwise known as Euler's Constant but must not be confused with Euler's number.

It was introduced by Euler in 1734.

In his 1790 work Adnotationes ad calculum integrale Euleri, Mascheroni published a calculation to 32 places of the value of this constant.

Only the first 19 places were accurate, but the remaining ones were corrected in 1809 by Johann von Soldner.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \gamma\)	\(:=\)	\(\displaystyle \lim_{n \to +\infty} \left({\sum_{k=1}^n \frac 1 k - \int_1^n \frac 1 x \ \mathrm dx}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{n \to +\infty} \left({H_n - \ln n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)