# Definition:Euler-Mascheroni Constant

## 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.

## Source of Name

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

## Historical Note

The Euler-Mascheroni Constant was introduced by Leonhard Paul Euler in $1734$.

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

1997: David Wells: Curious and Interesting Numbers (2nd ed.) claims that Euler calculated it to $16$ places at some point, but give no further details, and this has not yet been corroborated.

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.

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: 257 – 285), and this may indeed be the first.