Abstract： In this paper, we suggest one method of finding the Green tensor functions in whole space, which are defined in formula (3). Some times it is also called the elementary solution of the corresponding differential equation. All this method is based on Fourier transform. Owing to complexity, we are obliged to make some simplifications. Magneto-gyrotropic media and electric-gyrotropic media are considered separately. For magneto-gyrotropic medium, such as ferrite, μ, is a tensor while ε remains scalar. Conversely, for electric-gyrotropic medium, such as plasma, ε is tensor and μ remains scalar. Taking advantage of the smallness of matrix μp (defined in (15)) we make an expansion in power series of μp, and carry out the calculations in first order approximation. The concrete results are expressed in formulae (23)、(25)、(28)、(32) and (33). The physical meaning of Γ and the effective region of the asymptotic expansions are discussed.