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(...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1993-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171293000936 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832563967861456896 |
|---|---|
| author | E. K. Koh C. K. Li |
| author_facet | E. K. Koh C. K. Li |
| author_sort | E. K. Koh |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-247e030c6a63435cb4779e3a3b614614 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1993-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-247e030c6a63435cb4779e3a3b6146142025-02-03T01:12:05ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251993-01-0116474975410.1155/S0161171293000936On defining the generalized functions δα(z) and δn(x)E. K. Koh0C. K. Li1Department of Mathematics and Statistics, University of Regina, Regina S4S 0A2, CanadaDepartment of Mathematics and Statistics, University of Regina, Regina S4S 0A2, CanadaIn 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.http://dx.doi.org/10.1155/S0161171293000936the δ-functiongeneralized functionsHilbert integral. |
| spellingShingle | E. K. Koh C. K. Li On defining the generalized functions δα(z) and δn(x) International Journal of Mathematics and Mathematical Sciences the δ-function generalized functions Hilbert integral. |
| title | On defining the generalized functions δα(z) and δn(x) |
| title_full | On defining the generalized functions δα(z) and δn(x) |
| title_fullStr | On defining the generalized functions δα(z) and δn(x) |
| title_full_unstemmed | On defining the generalized functions δα(z) and δn(x) |
| title_short | On defining the generalized functions δα(z) and δn(x) |
| title_sort | on defining the generalized functions δα z and δn x |
| topic | the δ-function generalized functions Hilbert integral. |
| url | http://dx.doi.org/10.1155/S0161171293000936 |
| work_keys_str_mv | AT ekkoh ondefiningthegeneralizedfunctionsdazanddnx AT ckli ondefiningthegeneralizedfunctionsdazanddnx |