Fractional residues Aleksandrs Mihailovs Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 http://www.math.upenn.edu/~mihailov/ Invariants of generalized tensor fields on a line are classified using special polynomials P_mk^(-1/lambda) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is defined. Some applications to the geometric quantization of a line and conformal quantum field theory are discussed as well. Preprint (March 5, 1998)