International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 4, Pages 749-754
doi:10.1155/S0161171293000936
On defining the generalized functions δα(z) and δn(x)
Department of Mathematics and Statistics, University of Regina, Regina S4S 0A2, Canada
Received 20 April 1992; Revised 10 August 1992
Copyright © 1993 E. K. Koh and C. K. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.