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We have previously developed a polymer-like expansion that applies when the(effective) action in a functional integral is an analytic function of thefields being integrated. Here, we develop methods to aid the application ofthis technique when the method of steepest descent is used to analyze thefunctional integral. We develop a version of the Banach fixed point theoremthat can be used to construct and control the critical fields, as analyticfunctions of external fields, and substitution formulae to control the changein norms that occurs when one replaces the integration fields by the sum of thecritical fields and the fluctuation fields.

Power Series Representations for Complex Bosonic Effective Actions. III. Substitution and Fixed Point Equations