Method for Borel-summing instanton singularities: Introduction

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Abstract
The problem of Borel-summing the perturbation series of field theories possessing instanton solutions to the Euclidean equations of motion is studied. Such theories have a singularity on the positive real axis of the Borel plane along which the Borel integration is performed. A direct functional expression for the Borel transform of a functional integral is given, which yields a prescription for integration over such singular points. The problem of "double counting" perturbation expansions about multiple solutions to the Euclidean equations of motion is solved. In this first of two papers the method is introduced and applied to a trivial integral possessing an instanton. Also Borel singularities of nontrivial functional integrals not possessing instantons are found and compared to familiar large-order calculations.