Definition:Euler-Mascheroni Constant

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Definition

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

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

where $\sequence {H_n}$ are the harmonic numbers 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

The Euler-Mascheroni constant $\gamma$ is also known as Euler's constant.

However, this allows it to be confused with Euler's number, so its use is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.

It is also sometimes known as Mascheroni's constant.

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


Also see

  • Results about the Euler-Mascheroni constant can be found here.


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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