Definition:Euler-Mascheroni Constant

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Definition

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

\(\displaystyle \gamma\) \(:=\) \(\displaystyle \lim_{n \mathop \to +\infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \int_1^n \frac 1 x \rd x}\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to +\infty} \paren {H_n - \ln n}\)

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


Decimal Expansion

The decimal expansion of the Euler-Mascheroni constant $\gamma$ starts:

$\gamma \approx 0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \, 90082 \, 40243 \, 1 \ldots$


Also known as

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

It is also sometimes known as Mascheroni's constant.

Some sources denote it by the symbol $\mathrm C$.


Also see


Source of Name

This entry was named for Leonhard Paul Euler and Lorenzo Mascheroni.


Historical Note

The Euler-Mascheroni Constant was presented by Leonhard Paul Euler to the St. Petersburg Academy on $11$ March $1734$.

It was published in $1738$, calculated to $6$ decimal places, as $0 \cdotp 577218$ (although only the first $5$ were correct, as Euler himself surmised).

He subsequently calculated it to $16$ places in $1781$, and published this in $1785$.

Mascheroni published a calculation to $32$ places of the value of this constant.

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

In more modern times, Dura W. Sweeney published the results of the calculation of its value to $3566$ places in $1963$.


There exists disagreement over the question of who was first to name it $\gamma$ (gamma).

Some sources claim it was Euler who named it $\gamma$, in $1781$, while others suggest its first appearance of that symbol for it was in Lorenzo Mascheroni's $1790$ work Adnotationes ad calculum integrale Euleri.

However, a close study of those works indicates that Euler used $A$ and $C$, and in the work cited, Mascheroni used $A$ throughout.

An early appearance of the symbol $\gamma$ was by Carl Anton Bretschneider in his $1837$ paper Theoriae logarithmi integralis lineamenta nova (J. reine angew. Math. Vol. 17: pp. 257 – 285), and this may indeed be the first.


Sources

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