  In this paper we discuss in detail computational methods and new results for
one-loop virtual corrections to N = 4 super Yang-Mills scattering amplitudes
calculated to all orders in epsilon, the dimensional regularization parameter.
It is often the case that one-loop gauge theory computations are carried out to
order epsilon^0, since higher order in epsilon contributions vanish in the
small epsilon limit. We will show, however, that the higher order contributions
are actually quite useful. In the context of maximally supersymmetric
Yang-Mills, we consider two examples in detail to illustrate our point. First
we will concentrate on computations with gluonic external states and argue that
N = 4 supersymmetry implies a simple relation between all-orders-in-epsilon
one-loop N = 4 super Yang-Mills amplitudes and the first and second stringy
corrections to analogous tree-level superstring amplitudes. For our second
example we will derive a new result for the all-orders-in-epsilon one-loop
superamplitude for planar six-particle NMHV scattering, an object which allows
one to easily obtain six-point NMHV amplitudes with arbitrary external states.
We will then discuss the relevance of this computation to the evaluation of the
ratio of the planar two-loop six-point NMHV superamplitude to the planar
two-loop six-point MHV superamplitude, a quantity which is expected to have
remarkable properties and has been the subject of much recent investigation. To
make the presentation as self-contained as possible, we extensively review the
prerequisites necessary to understand the main results of this work.
