Using functional equations to calculate Feynman integrals

We propose a method for using functional equations to calculate Feynman integrals analytically. We describe the algorithm for solving the functional equations and show that a solution of a functional equation for the Feynman integral is a combination of several integrals with fewer kinematic variables. In some cases, using the functional equations, we can also reduce these integrals to integrals with even fewer variables. Such a stepwise application of the functional equations leads to integrals that can be calculated more simply than the original integral. We apply the proposed method to several one-loop integrals. For the three-point and four-point integrals with massless propagators and an arbitrary space dimension $d$, we obtain analytic expressions in terms of hypergeometric functions.