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. Our goal is to corroborate recent results for calculation of the local Fourier transforms and then extend our methods to a similar situation, the local Mellin transforms.

This dissertation is divided into three main parts. In the first part we prove explicit formulas for calculation of the local Fourier transforms. These formulas have recently been proved by others, and we reproduce their results using different techniques. The other two parts of the dissertation are given over to applying those same techniques to the local Mellin transforms for connections on the punctured formal disk. In the second part, 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. In the third part we find formulas for explicit calculation of the local Mellin transforms in the same spirit as the results for the local Fourier transforms.