Asymptotic representations of the amplitudes of the partial waves in the Mie series are obtained without any special functions. These representations are useful for description of the optical fields inside spherical aerosol particles and in the plasma spherical inhomogeneities. One can use the relationships obtained in the medium range of the diffraction parameter values, namely, for the close values of the product of the diffraction parameter by the complex refractive index and of the number of a partial wave. The comparative analysis is carried out of the amplitudes calculated by the asymptotic and Mie formulas. The asymptotic formulas obtained are most convenient for testing algorithms for calculating the amplitudes constructed using the exact Mie formulas.