# Definition:Euler-Mascheroni Constant

## 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 $H_n$ 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

- Existence of Euler-Mascheroni Constant where its existence is demonstrated.

- Euler's Integral Theorem, which proves that $H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$

## 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

- 1738: Leonhard Paul Euler:
*De Progressionibus Harmonicis Obseruationes*(*Commentarii Acad. Sci. Imp. Pet.***Vol. 7**: pp. 150 – 161)

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 1$: Special Constants: $1.20$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 14.3 \ (5)$ - 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,57721 56649 \ldots$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0 \cdotp 577 \, 215 \, 664 \, 901 \, 532 \, 860 \, 606 \, 512 \, 090 \, 082 \, 402 \, 431 \ldots$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \, 90082 \, 40243 \, 1 \ldots$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Euler's constant** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**harmonic series**