Dependence of dimensionless axial phase speed (ω/kv_Ai) on dimensionless axial wave number (kd) for the first several branches of proper kink eigenmodes. The density contrast is fixed at ρ_i/ρ_e = 5, whereas a number of steepness parameters μ are examined in different panels. The two horizontal dashed lines represent the internal and external Alfvén speeds (v_Ai and v_Ae). Different branches are distinguished by the line styles as illustrated in panel a, where our convention for labeling the eigenmodes is also shown. Cutoff axial wave numbers arise for all branches except the first one (j = 1), their labels chosen to be consistent with our mode labeling convention. Panel a singles out the dimensionless axial wave number kd = π/20 to illustrate the spectrum for the EVP presented in Sect. 3.1. The continuum sub-spectrum is represented by the vertical arrow, the eigenfrequencies ω continuously extending from kv_Ae to infinity. Only one element is present in the point sub-spectrum for this chosen kd, and it is represented by the asterisk.