The object is to study the linear stability of an unbounded stably stratified shear layer in an inviscid, Boussinesq fluid. The flow is modeled by the velocity and buoyancy frequency profiles: \(U=U_ 0 tanh(z/d)\) and \(N^ 2=N_ 1^ 2+N^ 2_ 2\mid tanh(z/d)\mid^{\alpha}\), where \(\alpha >0\). In three specific cases \((\alpha =0,2,4)\), the neutral modes are derived systematically using an analytical transform of the Taylor-Goldstein equation into the hypergeometric equation. Furthermore, the neutral modes, associated to propagating wave instabilities, correspond to gravity waves with infinite critical level reflection and transmission (i.e., resonant overreflection). It is to be noted that resonant overreflection is possible in the present model as long as the minimum Richardson number of the flow is smaller than 0.25. In the conclusion, the importance of the results obtained is discussed, in relation with the spontaneous generation of gravity waves in a stratified shear layer.