Fundamental Properties of Perturbation-Theoretical Integral Representations. III

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Abstract
Asymptotic behavior and subtraction problem of the perturbation‐theoretical integral representation (PTIR) are investigated in detail. Six theorems are rigorously proved in this connection. It is shown that a function represented by an unsubtracted PTIR may asymptotically increase in particular directions. The relation between the asymptotic behavior and the subtraction number is clarified for the subtracted PTIR. As a by‐product one obtains a consistent definition of a finite part of the integral involving x−1θ(x).