Important insights into point-source radiation in attenuative anisotropic media can be gained by applying asymptotic methods. Analytic description of point-source radiation can help in the interpretation of the amplitude-variation-with-offset response and physical-modeling data. We derive the asymptotic Green’s function in homogeneous, attenuative, arbitrarily anisotropic media using the steepest-descent method. The saddle-point condition helps describe the behavior of the slowness and group-velocity vectors of the far-field P-wave. Numerical results from the asymptotic analysis compare well with those obtained by the ray-perturbation method for P-waves in transversely isotropic media.