# 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: )
• $- c_k \le \gamma -1 + \ln(k) \sum_{l=1}^{12k+1} d(k,l) - \sum_{l=1}^{12k+1} \frac{d(k,l)}{l+1} \le c_k$, 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: )

Sources:

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

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