 1/"x" is the distribution obtained by the Cauchy principal value

for all "formula_1" ∈ "S"(R), and "δ" is the Dirac delta distribution then

but

so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions.Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone.In the context of quantum field theory, however, solutions can be found.