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Euler–Mascheroni constant
From OeisWiki
(Redirected from Harmonic sequence)
γ |
γ |
e |
It is defined as the limiting difference between the harmonic series and the natural logarithm, i.e.
⌊ x⌋ |
{ x } := x − ⌊ x⌋ |
x |
x ≥ 0 |
hence
γ |
Contents
Decimal expansion
The decimal expansion of the Euler–Mascheroni constant is
|
γ =
|
A001620 Decimal expansion of Euler's constant (or Euler–Mascheroni constant) gamma.
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{5, 7, 7, 2, 1, 5, 6, 6, 4, 9, 0, 1, 5, 3, 2, 8, 6, 0, 6, 0, 6, 5, 1, 2, 0, 9, 0, 0, 8, 2, 4, 0, 2, 4, 3, 1, 0, 4, 2, 1, 5, 9, 3, 3, 5, 9, 3, 9, 9, 2, 3, 5, 9, 8, 8, 0, 5, 7, 6, 7, 2, 3, 4, 8, 8, 4, 8, 6, 7, 7, 2, 6, 7, 7, 7, 6, 6, 4, ...}
The value is available in Pari/GP as "Euler" and WolframAlpha as "EulerGamma".
Continued fraction expansion
The simple continued fraction expansion of the Euler–Mascheroni constant is
γ |
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{0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, 11, 3, 7, 1, 7, 1, 1, 5, 1, 49, 4, 1, 65, 1, 4, 7, 11, 1, 399, 2, 1, 3, 2, 1, 2, 1, 5, 3, 2, 1, 10, 1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 1, 1, 2, 5, 1, 3, 6, 2, 1, 2, 1, 1, ...}
Square of the Euler–Mascheroni constant
The decimal expansion of the square of the Euler–Mascheroni constant is
|
γ 2 =
|
A155969 Decimal expansion of the square of the Euler–Mascheroni constant.
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{3, 3, 3, 1, 7, 7, 9, 2, 3, 8, 0, 7, 7, 1, 8, 6, 7, 4, 3, 1, 8, 3, 7, 6, 1, 3, 6, 3, 5, 5, 2, 4, 4, 2, 2, 6, 6, 5, 9, 4, 1, 7, 1, 4, 0, 2, 4, 9, 6, 2, 9, 7, 4, 3, 1, 5, 0, 8, 3, 3, 3, 3, 8, 0, 0, 2, 2, 6, 5, 7, 9, 3, 6, 9, 5, 7, 5, 6, 6, ...}
Reciprocal
The decimal expansion of the reciprocal of the Euler–Mascheroni constant is
√ 3 = 1.732050807568877… |
1 / γ = √ 3 × 1.0002331958335… |
A098907 Decimal expansion of
1 / γ |
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{1, 7, 3, 2, 4, 5, 4, 7, 1, 4, 6, 0, 0, 6, 3, 3, 4, 7, 3, 5, 8, 3, 0, 2, 5, 3, 1, 5, 8, 6, 0, 8, 2, 9, 6, 8, 1, 1, 5, 5, 7, 7, 6, 5, 5, 2, 2, 6, 6, 8, 0, 5, 0, 2, 2, 0, 4, 8, 4, 3, 6, 1, 3, 2, 8, 7, 0, 6, 5, 5, 3, 1, 4, 0, 8, 6, 5, 5, 2, ...}
Laurent expansion of the Riemann zeta function
From the Laurent expansion of the Riemann zeta function abouts = 1 |
See also
- OEIS format for decimal representation of constants
- Stieltjes constants (sometimes referred to as generalized Euler constants)
- Meissel–Mertens constant (the analogue of Euler–Mascheroni constant for the harmonic series of the primes)
Notes
- ↑ Štefan Porubský: Euler-Mascheroni Constant. Retrieved 2012/9/20 from Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, web-page http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/MathematicalConstants/EulerMascheroni.htm.
- ↑ Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource.
- ↑ John Albert, Some unsolved problems in number theory, Department of Mathematics, University of Oklahoma.
External links
- Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource.