\textbf{\textit{v}}$ and $ d \textbf{\textit{x}}$ denote the elements of velocity and position vectors.

@Z76 used Kalnajs’s theory to compute the modes of the isothermal disk [@M63], which has the astrophysically important property of a flat rotation curve. His analysis was then extended by @ER98a [@ER98b] to general scale-free disks with arbitrary cusp slopes. Application of Kalnajs’s theory to soft-centered models of stellar disks has been mainly focused on the isochrone and Kuzmin-Toomre disks [@K78; @H92; @PC97]. A disk with exponential light profile and an approximately flat rotation curve was also investigated by @VD96. More recently Jalali & Hunter (2005a, hereafter JH) gave new results for soft-centered models of stellar disks. They showed the importance of a boundary integral in the modal properties of unidirectional disks and computed a fundamental bar mode and a secondary spiral mode for the isochrone, Kuzmin-Toomre and a newly introduced family of cored exponential disks. JH also extended Kalnajs’s first order perturbation theory to the second order, and illustrated energy and angular momentum content of different Fourier components. Their bar charts showed that only a few number of expansion terms in the radial angle govern the perturbed dynamics.

Implementation of Kalnajs’s (1977) theory, however, has some 