How to (Path-) Integrate by Differentiating

TL;DR

This paper introduces novel differentiation-based techniques for expressing integrals, including Laplace transforms and path integrals, enhancing analytical tools and perturbative methods in mathematical physics.

Contribution

It extends differentiation-based integral representations to Laplace transforms, inverse transforms, and path integrals, offering new approaches for analytical and perturbative calculations.

Findings

Expressed Laplace transform via derivatives

Derived inverse Laplace transform using differentiation

Applied methods to path integrals in quantum field theory

Abstract

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 Laplace 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.