The linearized Vlasov–Maxwell equations are used to investigate harmonic stability properties for a planar wiggler free‐electron laser (FEL). The analysis is carried out in the Compton regime for a tenuous electron beam propagating in thezdirection through the constant‐amplitude planar wiggler magnetic fieldB0=−Bw cos k0zeˆx. Transverse spatial variations are neglected (∂/∂x=0=∂/∂y), and the case of an FEL oscillator (temporal growth) is considered. Assuming ultrarelativistic electrons and &kgr;2=a2w/(&ggr;20−1) ≪1, wherea2w=e2B2w/m2c4k20and &ggr;0mc2is the electron energy, the kinetic dispersion relation is derived in the diagonal approximation for perturbations about general beam equilibrium distribution functionG+0(&ggr;0). Because of the wiggler modulation of the axial electron orbits, strong wave–particle interaction can occur for &ohgr;≊[k+k0(1+2l)] &bgr;Fc, where &bgr;Fcis the axial velocity, &ohgr; andkare the wave oscillation frequency and wavenumber, respectively, andl=0, 1, 2, . . . are harmonic numbers corresponding to an upshift in frequency. The strength of thelth harmonic wave–particle coupling is proportional toKl(b1) =[Jl(b1)−Jl+1(b1)]2, whereb1=(k/8k0)&kgr;2. Assuming thatG+0(&ggr;0) is strongly peaked around &ggr;0=&ggr;ˆ≫1, detailedlth harmonic stability properties are investigated for (a) strong FEL instability corresponding to monoenergetic electrons (&Dgr;&ggr;=0), and (b) weak resonant FEL instability corresponding to a sufficiently large energy spread that ‖Im &ohgr;/[k+k0(1+2l)] &Dgr;vz‖≪1.For monoenergetic electrons the characteristic maximum growth rate scales as [Kl(bˆ1)(1+2l)]1/3, which exhibits a relatively weak dependence on harmonic numberl. Here,bˆ1= 1/2 [a2w/(2+a2w)] (1+2l). On the other hand, for weak resonant FEL instability, the growth rate scales asKl(bˆ1)/(1+2l), which decreases rapidly for harmonic numbersl≥1.