Eikonal Blog

2010.01.04

Euler-Mascheroni constant

Filed under: mathematics — Tags: , — sandokan65 @ 19:52
  • \gamma :\equiv \lim_{n\rightarrow \infty} (\sum_{k=1}^n \frac1{k} - \ln(n+1)).
  • \gamma \sim 0.577,215,664,9...
  • \gamma = - \int_0^\infty e^{-t} \ln(t) dt.

For the more precise numerical value see https://eikonal.wordpress.com/2010/03/09/eulers-constant-euler-mascheroni-constant-gamma/.

Also:

  • \psi(1) = \Gamma'(1) = -\gamma,
  • \psi(1/2) = - \gamma -2\ln(2),
  • \gamma = 1 - \Gamma'(2).

Inequalities:

  • \frac1{24 (n+1)^2} < R_n - \gamma <  \frac1{24 n^2}, where R_n :\equiv \sum_{k=1}^n \frac1{k} - \ln(n+\frac12)). (source: [2])
  • - c_k \le \gamma -1 + \ln(k) \sum_{l=1}^{12k+1} d(k,l) - \sum_{l=1}^{12k+1} \frac{d(k,l)}{l+1} \le c_k, where c_k :\equiv \frac{2}{(12k)!} + 2k^2 e^{-k} and d(k,l) = (-)^{l-1} \frac{k^{l+1}}{(l-1)!(l+1}. (source: [2])

Sources:

  1. [1] “The Functions of Mathematical Physics” by Harry Hochstadt
  2. [2] “Topics in Special Functions” by G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen – http://arxiv.org/abs/0712.3856

1 Comment »

  1. […] by Euler-Mascheroni constant « Eikonal Blog — 2010.03.16 @ 11:47 […]

    Like

    Pingback by Transcendental numbers: Euler’s constant (= Euler-Mascheroni constant) gamma « Eikonal Blog — 2010.03.16 @ 11:50


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