  This paper provides specific results on the Igusa local zeta function for the
curves $x^n+y^m$. In addition to specific results, we give an introduction to
$p$-adic analysis and a discussion of various methods which have been used to
compute these zeta functions. The paper was written by the 1992 REU group in
$p$-adic analysis supervised by Margaret Robinson. It has been available on the
Mount Holyoke REU website.
