  We calculate gauge theory one-loop amplitudes with the aid of the complex
shift used in the Britto-Cachazo-Feng-Witten (BCFW) recursion relations of tree
amplitudes. We apply the shift to the integrand and show that the contribution
from the limit of infinite shift vanishes after integrating over the loop
momentum, by a judicious choice of basis for polarization vectors. This enables
us to write the one-loop amplitude in terms of on-shell tree and lower point
one-loop amplitudes. Some of the tree amplitudes are forward amplitudes. We
show that their potential singularities do not contribute and the BCFW
recursion relations can be applied in such a way as to avoid these
singularities altogether. We calculate in detail $n$-point one-loop amplitudes
for $n=2,3,4$, and outline the generalization of our method to $n>4$.
