  Physical processes ranging from the Lamb shift to the energy loss dE/dx of a
charged particle traversing a plasma entail processes that occur over a wide
range of energy or length scales. Different physical mechanisms dominate at one
or the other end of this range. For example, in the energy loss problem, soft
collisions that are screened by collective effects are important at large
distances, while at short distances hard collisions are important where the
exact details of the single-particle interactions must be taken into account.
We introduce a novel application of dimensional continuation. The soft
processes dominate at all scales when the spatial dimension \nu is less than 3,
and we use them to compute the result to leading order for \nu < 3. On the
other hand, the hard processes dominate at all scales for \nu > 3, and we use
them to compute the result to leading order for these spatial dimensions. We
then explain why the sum of the analytic continuation of these disparate
mechanisms yields the correct leading-order result for the physical limit at
\nu = 3 dimensions. After applying this new method to the energy loss problem
in some detail, we then show how it also provides a very short and easy way to
compute the Lamb shift.
