Approximation of Sondow’s generalized-Euler-constant function on the interval [−1, 1]

Abstract

Using the Euler-Maclaurin (Boole/Hermite) summation formula, the generalized-Euler-Sondow-constant function γ(z),

$$ \gamma(z):=\sum_{k=1}^{\infty}z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right) \qquad (-1\le z\le 1),$$

where ${\gamma(-1)=\ln\frac{4}{\pi}}$ and γ(1) is the Euler-Mascheroni constant, is estimated accurately.