Explicit Expression of Foldy-Wouthuysen Transformation

The general, explicit and formally closed expression of arbitrary n-times Foldy-Wouthuysen transformations is clearly and strictly derived out. It is proved that if transformed Hamiltonian needs to be approximated to the order 1/m^K or mv^2K when to involve the orders of the operators, then to make N = (K + 1)/2-times Foldy-Wouthuysen transformations is just enough ( “...”means to take the part of integer). An example in non-relativistic quantum chromodynamics is given.