  Within the framework of Soft Collinear Effective Theory, we present
calculations of semi-inclusive jet functions and fragmenting jet functions at
next-to-leading order (NLO) for both quark- and gluon-initiated jets, for jet
algorithms of $J_{E_T}^{(I)}$ and $J_{E_T}^{(II)}$ where one maximizes a
suitable jet function. We demonstrate the consistency of the obtained results
with the standard perturbative QCD calculations for $J_{E_T}^{(I)}$ algorithm,
while the results for fragmenting jet functions with the $J_{E_T}^{(II)}$
algorithm are new. The renormalization group (RG) equation for both
semi-inclusive jet functions and fragmenting jet functions are derived and
shown to follow the time-like DGLAP evolution equations, independent of
specific jet algorithms. The RG equation can be used to resum single logarithms
of the jet size parameter $\beta$ for highly collimated jets in these
algorithms where $\beta \gg 1$.
