  Much recent work has been done on the local Fourier transforms for
connections on the punctured formal disk. Specifically, the local Fourier
transforms have been introduced, shown to induce certain equivalences of
categories, and explicit formulas have been found to calculate them. In this
paper, we prove analogous results in a similar situation, the local Mellin
transforms for connections on the punctured formal disk. Specifically, we
introduce the local Mellin transforms and show that they induce equivalences
between certain categories of vector spaces with connection and vector spaces
with invertible difference operators, as well as find formulas for explicit
calculation in the same spirit as the calculations for the local Fourier
transforms.
