On defining the generalized functions δα(z) and δn(x)

Abstract

In a previous paper (see [5]), we applied a fixed δ-sequence and neutrix limit due to Van der Corput to give meaning to distributions δk and (δ′)k for k∈(0,1) and k=2,3,…. In this paper, we choose a fixed analytic branch such that zα(−π<argz≤π) is an analytic single-valued function and define δα(z) on a suitable function space Ia. We show that δα(z)∈I′a. Similar results on (δ(m)(z))α are obtained. Finally, we use the Hilbert integral φ(z)=1πi∫−∞+∞φ(t)t−zdt where φ(t)∈D(R), to redefine δn(x) as a boundary value of δn(z−i ϵ ). The definition of δn(x) is independent of the choice of δ-sequence.