Recently some doubt has arisen about the accuracy of a finite difference approach used in the linear stability analysis of velocity profiles representative of tropical cyclones. There are conflicting results in the literature concerning the stability of the azimuthal wavenumber-one mode, based on the f-plane non-divergent barotropic vorticity equation in cylindrical coordinates. In this paper it is shown via finite difference codes that this mode is stable, for five such velocity profiles. A better representation of the outer boundary condition also yields more accurate growth rates. A tighter semicircle bound than that used in some shooting methods is derived, incorporating wavenumber dependence. A global matrix (“pseudo-spectral”) method, the discrete ordinate method, also appears to work remarkably well—especially for lower unstable wavenumbers at coarse domain resolutions.