Closed-Form Solutions of Zero Dimensional $\phi^4$-Field Theory Using Bessel Functions: A Non-Perturbative Approach

The integral $\int_{-\infty}^{\infty} e^{- x^2 - g x^4} dx$ is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically it is analysed via perturbation theory. Close form solutions have been quoted but it is not clear how they were derived. So I set about deriving the close form solution on my own and using the same methodology obtain closed form expressions for the even positive integer moments.