The resonant excitation of Alfv¿n waves is considered in a cold plasma embedded in a uniform magnetic field Bzˆ. All wave fields are assumed to vary as exp[i(&lgr;y-&ohgr;t)>, and the background medium is invariant in y. The background density distribution &rgr;0(x,z) is otherwise completely arbitrary. Regular and singular solutions for the waves are derived systematically in the vicinity of a resonance by considering a generalized Frobenius series, and we are able to recover many results found in earlier studies. Some new features of our work include a generalization of the overlap integral determining the efficiency with which any resonance may be excited, and the possibility that &xgr;y∝1/xN (N=1,2,3...) at the resonance. Hitherto only the solution with N=1 has been considered. ¿ American Geophysical Union 1993