Suppose one was trying to find out whether a function has exactly one non-removable singularity in a given region on the complex plane or not, but for some reason could not find out directly. Does it suffice to take a closed path integral around the boundary of the region, and see if it equals zero? That is, if one determines that the path integral around the region's boundary is zero, does one know for sure that there isn't exactly one singularity? Or, vice versa, if one finds out that the integral isn't zero, does one automatically know that there's at least one singularity? Thanks for the help.