Derivation of Bessel function closed-form solutions in zero dimensional φ4-field theory

Abstract

The integral ∫−∞∞e−xdx is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically, it is analyzed via perturbation theory. Closed-form solutions have been quoted for which I could not find any derivation. Using a simple and elegant transformation, the close form solutions for the integral and its even positive integer moments can be obtained in terms of Bessel functions.