Left: damping integral ℐ⁵(z,X) as defined in Eq. (B.5)for retrograde IGW (m< 0) and as a function of X = ω/mΔΩ (solid lines). The dashed lines represent the approximate expressions given in Eqs. (B.10) and (B.11). Right: damping integral ℐ⁵(z,X) for prograde IGW (m> 0) as a function of (solid lines). The dashed and the dashed-dotted lines respectively represent the approximate expressions at high and low values for as given in Eq. (B.12). Both panels: two values of z = r/r_t are considered, r/r_t = 0.9 (blue color) and r/r_t = 0.1 (green color). The vertical dotted lines indicates the values at the transition of regime, X_t, as defined in Appendix B.1.2.