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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Natural Sciences Publishing
2012
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| Online Access: | http://psasir.upm.edu.my/id/eprint/25279/1/Some%20results%20on%20the%20gamma%20function%20for%20negative%20integers.pdf http://psasir.upm.edu.my/id/eprint/25279/ http://www.naturalspublishing.com/Article.asp?ArtcID=425 |
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| Summary: | 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. |
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