Fourier-Feynman transforms and the first variation

In this paper we complete the following four objectives: 1. We obtain an integration by parts formula for analytic Feynman integrals. 2. We obtain an integration by parts formula for Fourier-Feynman transforms. 3. We find the Fourier-Feynman transform of a functionalF from a Banach algebra after it has been multiplied byn linear factors. 4. We evaluate the analytic Feynman integral of functionals like those described in 3 above. A very fundamental result by Cameron and Storvick 5, Theorem 1], in which they express the analytic Feynman integral of the first variation of a functionalF in terms of the analytic Feynman integral ofF multiplied by a linear factor, plays a key role throughout this paper.