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We study the virial expansion for three-dimensional Bose and Fermi gases atfinite temperature using an approximation that only considers two-bodyprocesses and is valid for high temperatures and low densities. The firstvirial coefficients are computed and the second is exact. The results areobtained for the full range of values of the scattering length and the unitarylimit is recovered as a particular case. A weak coupling expansion is performedand the free case is also obtained as a proper limit. The influence of ananisotropic harmonic trap is considered using the Local Density Approximation -LDA, analytical results are obtained and the special case of the isotropic trapis discussed in detail.

Virial coefficients for Bose and Fermi trapped gases beyond the unitary limit: an S-Matrix approach