Examples of exact versus approximate dispersion relations. [Top-left] real frequency and [top-right] growth rate for cyclotron instability (case 30). Solid curves are exact numerical solutions based upon equation (A.1), while the dashed curves are approximate solutions, equations (16) and (18), except that z_i is multiplied by the scaling factor f = 0.305. [Bottom-left] Real frequency and [bottom-right] growth rate for firehose instability (case 43). Solid curves are exact numerical solutions based upon equation (A.1), while the dashed curves are approximate solutions based on equations (17) and (19), except that z_i is multiplied by f = 1.35.