  We discuss the current use of the operator product expansion in QCD
calculations. Treating the OPE as an expansion in inverse powers of an
energy-squared variable (with possible exponential terms added), approximating
the vacuum expectation value of the operator product by several terms and
assuming a bound on the remainder along the euclidean region, we observe how
the bound varies with increasing deflection from the euclidean ray down to the
cut (Minkowski region). We argue that the assumption that the remainder is
constant for all angles in the cut complex plane is not justified. Making
specific assumptions on the properties of the expanded function, we obtain
bounds on the remainder in explicit form and show that they are very sensitive
both to the deflection angle and to the class of functions chosen. The results
obtained are discussed in connetcion with calculations of the coupling constant
\alpha_{s} from the \tau decay.
