On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse 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 cert...
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| Format: | Article |
| Language: | English English |
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Hindawi Publishing Corporation
2011
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| Online Access: | http://psasir.upm.edu.my/id/eprint/25270/ http://psasir.upm.edu.my/id/eprint/25270/1/On%20the%20composition%20and%20neutrix%20composition%20of%20the%20delta%20function%20with%20the%20hyperbolic%20tangent%20and%20its%20inverse%20functions.pdf |
| _version_ | 1848845263295217664 |
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| author | Fisher, Brian Kilicman, Adem |
| author_facet | Fisher, Brian Kilicman, Adem |
| author_sort | Fisher, Brian |
| building | UPM Institutional Repository |
| collection | Online Access |
| 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. It is proved that the neutrix composition δ(rs-1) ((tanh x+)1/r) exists and δ(rs-1) ((tanh x+)1/r) =√k=0 s-1√i=0 Kk ((- 1)k cs-2 i - 1, k (rs) !/2sk!) δ(k) (x) for r, s = 1,2,⋯ , where Kk is the integer part of (s - k - 1) / 2 and the constants c j,k are defined by the expansion (tanh - 1 x)k = {√i=0 ∞ (x 2i+1/(2 i + 1)) }k = √j=k ∞ c j, k xj, for k = 0,1, 2,⋯. Further results are also proved. |
| first_indexed | 2025-11-15T08:44:02Z |
| format | Article |
| id | upm-25270 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T08:44:02Z |
| publishDate | 2011 |
| publisher | Hindawi Publishing Corporation |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-252702015-09-21T04:42:14Z http://psasir.upm.edu.my/id/eprint/25270/ On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. Fisher, Brian Kilicman, Adem 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. It is proved that the neutrix composition δ(rs-1) ((tanh x+)1/r) exists and δ(rs-1) ((tanh x+)1/r) =√k=0 s-1√i=0 Kk ((- 1)k cs-2 i - 1, k (rs) !/2sk!) δ(k) (x) for r, s = 1,2,⋯ , where Kk is the integer part of (s - k - 1) / 2 and the constants c j,k are defined by the expansion (tanh - 1 x)k = {√i=0 ∞ (x 2i+1/(2 i + 1)) }k = √j=k ∞ c j, k xj, for k = 0,1, 2,⋯. Further results are also proved. Hindawi Publishing Corporation 2011 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/25270/1/On%20the%20composition%20and%20neutrix%20composition%20of%20the%20delta%20function%20with%20the%20hyperbolic%20tangent%20and%20its%20inverse%20functions.pdf Fisher, Brian and Kilicman, Adem (2011) On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. Journal of Applied Mathematics, 2011 (846736). pp. 1-14. ISSN 1110-757X; ESSN:1687-0042 10.1155/2011/846736 English |
| spellingShingle | Fisher, Brian Kilicman, Adem On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title_full | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title_fullStr | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title_full_unstemmed | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title_short | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| title_sort | on the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions. |
| url | http://psasir.upm.edu.my/id/eprint/25270/ http://psasir.upm.edu.my/id/eprint/25270/ http://psasir.upm.edu.my/id/eprint/25270/1/On%20the%20composition%20and%20neutrix%20composition%20of%20the%20delta%20function%20with%20the%20hyperbolic%20tangent%20and%20its%20inverse%20functions.pdf |