Differential forms of the dispersion integral

Abstract

We discuss the derivative analyticity relations which were originally proposed as an alternative to dispersion relations; the dispersion integral is replaced by a tangent series of derivatives of the functionf which is, in the majority of high-energy applications, the imaginary part of a scattering amplitude. We consider three ways how to give the tangent series precise meaning. If the series converges on a real interval,f must be extensible to an entire function. Iff ∈ C∞ and the dispersion integral converges, the latter is equal to a generalized sum of the tangent series. Finally in the high-energy limit, derivative relations are valid in which the tangent series is replaced by its first term. Then the class of applicability includes the majority of physically interesting functions.