Some results on the gamma function for negative integers
The Gamma function Γ (s)(-r) is defined by Γ (s)(-r) = N - lim ε→0 ∫ε ∞ t -r-1, ln s t e -t dt for r, s = 0, 1, 2, . . . , where N is the neutrix having domain N′ = {ε : 0 < ε < ∞} with negligible functions finite linear sums of the functions ε λ ln s-1 ε, ln s ε : λ < 0, s = 1, 2,. .. and...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Natural Sciences Publishing
2012
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| Online Access: | http://psasir.upm.edu.my/id/eprint/25279/ http://psasir.upm.edu.my/id/eprint/25279/1/Some%20results%20on%20the%20gamma%20function%20for%20negative%20integers.pdf |
| _version_ | 1848845266064506880 |
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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 | The Gamma function Γ (s)(-r) is defined by Γ (s)(-r) = N - lim ε→0 ∫ε ∞ t -r-1, ln s t e -t dt for r, s = 0, 1, 2, . . . , where N is the neutrix having domain N′ = {ε : 0 < ε < ∞} with negligible functions finite linear sums of the functions ε λ ln s-1 ε, ln s ε : λ < 0, s = 1, 2,. .. and all functions which converge to zero in the normal sense as CMMI9.-1.epsilon1 tends to zero. In the classical sense Gamma functions is not defined for the negative integer. In this study, it is proved that for r = 1, 2,..., where φ(r) = Σ r i=1 1/i. Further results are also proved. |
| first_indexed | 2025-11-15T08:44:05Z |
| format | Article |
| id | upm-25279 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T08:44:05Z |
| publishDate | 2012 |
| publisher | Natural Sciences Publishing |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-252792017-10-26T10:10:26Z http://psasir.upm.edu.my/id/eprint/25279/ Some results on the gamma function for negative integers Fisher, Brian Kilicman, Adem The Gamma function Γ (s)(-r) is defined by Γ (s)(-r) = N - lim ε→0 ∫ε ∞ t -r-1, ln s t e -t dt for r, s = 0, 1, 2, . . . , where N is the neutrix having domain N′ = {ε : 0 < ε < ∞} with negligible functions finite linear sums of the functions ε λ ln s-1 ε, ln s ε : λ < 0, s = 1, 2,. .. and all functions which converge to zero in the normal sense as CMMI9.-1.epsilon1 tends to zero. In the classical sense Gamma functions is not defined for the negative integer. In this study, it is proved that for r = 1, 2,..., where φ(r) = Σ r i=1 1/i. Further results are also proved. Natural Sciences Publishing 2012 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/25279/1/Some%20results%20on%20the%20gamma%20function%20for%20negative%20integers.pdf Fisher, Brian and Kilicman, Adem (2012) Some results on the gamma function for negative integers. Applied Mathematics & Information Sciences, 6 (2). pp. 173-176. ISSN 1935-0090; ESSN: 2325-0399 http://www.naturalspublishing.com/Article.asp?ArtcID=425 |
| spellingShingle | Fisher, Brian Kilicman, Adem Some results on the gamma function for negative integers |
| title | Some results on the gamma function for negative integers |
| title_full | Some results on the gamma function for negative integers |
| title_fullStr | Some results on the gamma function for negative integers |
| title_full_unstemmed | Some results on the gamma function for negative integers |
| title_short | Some results on the gamma function for negative integers |
| title_sort | some results on the gamma function for negative integers |
| url | http://psasir.upm.edu.my/id/eprint/25279/ http://psasir.upm.edu.my/id/eprint/25279/ http://psasir.upm.edu.my/id/eprint/25279/1/Some%20results%20on%20the%20gamma%20function%20for%20negative%20integers.pdf |