Normal forms of parabolic logarithmic transseries

We give formal normal forms for parabolic logarithmic transseries $f=z+\cdots \,$, with respect to parabolic logarithmic normalizations. Normalizations are given algorithmically, using fixed point theorems, as limits of Picard's sequences in appropriate complete metric spaces, in contrast to transfinite \emph{term-by-term} eliminations described in former works. Furthermore, we give the explicit formula for the residual coefficient in the normal form and show that, in the larger logarithmic class, we can even eliminate the residual term from the normal form.