  In this paper, we study power series having a fixed point of multiplier 1.
First, we give a closed formula for the residue fixed point index, in terms of
the first coefficients of the power series. Then, we use this formula to study
wildly ramified power series in positive characteristic. Among power series
having a multiple fixed point of small multiplicity, we characterize those
having the smallest possible lower ramification numbers in terms of the residue
fixed point index. Furthermore, we show that these power series form a generic
set, and, in the case of convergent power series, we also give an optimal lower
bound for the distance to other periodic points.
