The Euclidean loop expansion for massive lambda PHI44: through one loop

As an application of the theory of solutions of the classical, Euclidean field equation, we prove the existence of solutions to the renormalized functional field equation, for the lambda PHI4 interaction in four Euclidean space dimensions, with non-negative lambda and nonzero mass, through order h/2πc. That is, we prove that the functional derivative of the connected generating functional is in the Schwartz space ReS(R4), when evaluated at external sources in ReS, through order h/2πc. We also prove the existence of all functional derivatives of the connected generating functional through the same order. All quantities of interest are analytic in the coupling constant at 0 <= lambda < infinitely, and continuous in the external source. (orig.)