On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions
Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequenc...
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| Language: | English |
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Wiley
2011-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2011/612353 |
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| author | Brian Fisher Adem Kılıçman |
| author_facet | Brian Fisher Adem Kılıçman |
| author_sort | Brian Fisher |
| collection | DOAJ |
| description | Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh-1x+)r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh-1x+)r] exists and is given by δ(s)[(sinh-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), for s=0,1,2,… and r=1,2,…, where cs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved. |
| format | Article |
| id | doaj-art-b82d4dbb586b45b381b3a3c5ed35c4c7 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b82d4dbb586b45b381b3a3c5ed35c4c72025-02-03T00:59:31ZengWileyJournal of Applied Mathematics1110-757X1687-00422011-01-01201110.1155/2011/612353612353On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine FunctionsBrian Fisher0Adem Kılıçman1Department of Mathematics, University of Leicester, Leicester LE1 7RH, UKDepartment of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, MalaysiaLet F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh-1x+)r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh-1x+)r] exists and is given by δ(s)[(sinh-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), for s=0,1,2,… and r=1,2,…, where cs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.http://dx.doi.org/10.1155/2011/612353 |
| spellingShingle | Brian Fisher Adem Kılıçman On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions Journal of Applied Mathematics |
| title | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |
| title_full | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |
| title_fullStr | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |
| title_full_unstemmed | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |
| title_short | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |
| title_sort | on the neutrix composition of the delta and inverse hyperbolic sine functions |
| url | http://dx.doi.org/10.1155/2011/612353 |
| work_keys_str_mv | AT brianfisher ontheneutrixcompositionofthedeltaandinversehyperbolicsinefunctions AT ademkılıcman ontheneutrixcompositionofthedeltaandinversehyperbolicsinefunctions |