On defining the generalized functions δα(z) and δn(x)
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(...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1993-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171293000936 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |