Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier and Lapla
ce transforms. The new methods also yield new perturbative expansions when the integrals cannot be solved analytically. Here, we add new results, for example, on expressing the Laplace transform and its inverse in terms of derivatives. The new methods can be used to express path integrals in terms of functional differentiation, and they also suggest new perturbative expansions in quantum field theory.
We derive new all-purpose methods that involve the Dirac Delta distribution. Some of the new methods use derivatives in the argument of the Dirac Delta. We highlight potential avenues for applications to quantum field theory and we also exhibit a con
nection to the problem of blurring/deblurring in signal processing. We find that blurring, which can be thought of as a result of multi-path evolution, is, in Euclidean quantum field theory without spontaneous symmetry breaking, the strong coupling dual of the usual small coupling expansion in terms of the sum over Feynman graphs.
