  We study a classification of the kappa-times integrated semigroups (for
kappa>0) by the (uniform) rate of convergence at the origin:
$\|S(t)\|=O(t^\alpha)$, $0\leq\alpha\leq\kappa$. By an improved generation
theorem we characterize this behaviour by Hille-Yosida type estimates. Then we
consider integrated semigroups with holomorphic extension and characterize the
convergence at the origin, as well as the existence of boundary values, by
estimates of the associated holomorphic semigroup. Different examples
illustrate these results. The particular case $\alpha=\kappa$, which
corresponds to the notions of Riesz means or tempered integrated semigroups, is
of special interest : as an application, it leads to an integrated version of
Euler's exponential formula.
