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 $\sequence {H_n}$ 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
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 $H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
- 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
- 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): $0 \cdotp 577 \, 215 \, 664 \, 901 \, 532 \, 860 \, 606 \, 512 \, 090 \, 082 \, 402 \, 431 \ldots$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Euler-Mascheroni constant
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Euler's constant
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Mascheroni's constant
- 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