Definition:Euler-Mascheroni Constant
Contents
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:
- $\gamma \approx 0 \cdotp 57721 \, 56649 \, 01532 \, 86060 \, 65120 \, 90082 \, 40243 \, 1 \ldots$
This sequence is A001620 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
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.
Also see
- Existence of Euler-Mascheroni Constant where its existence is demonstrated.
Sources
- 1738: Leonhard Paul Euler: De Progressionibus Harmonicis Obseruationes (Commentarii Acad. Sci. Imp. Pet. Vol. 7: 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): $\S 14.3 \ (5)$
- 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 ($1707$ – $1783$)
- 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
