Extended x-ray-absorption fine-structure (EXAFS) of copper has been measured from 4 to 500 K and analyzed by the cumulant method, to check the effectiveness of EXAFS as a probe of local dynamics and thermal expansion. The comparison between parallel mean square relative displacements (MSRD) of the first four coordination shells has allowed detecting a significant deviation from a pure Debye behavior. The first- shell EXAFS thermal expansion is larger than the crystallographic one: the difference has allowed evaluating the perpendicular MSRD, whose Debye temperature is slightly larger than the one of the parallel MSRD, due to anisotropy effects. High-order first-shell cumulants are in good agreement with quantum perturbative mod- els. The anharmonic contribution to the first-shell parallel MSRD amounts to less than 1.5 percent. The third cumulant cannot be neglected in the analysis, if accurate values of the first cumulant are sought; it cannot however be used to directly estimate the thermal expansion. The shape of the effective pair potential is independent of temperature; a rigid shift, partially due to the relative motion perpendicular to the bond direction, is however observed.