Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article Citation - WoS: 1Citation - Scopus: 1On the Non-Commutative Neutrix Product of the Distributions X<sup>-r</Sup>+ Ln<sup>p</Sup> X+ and X<sup>μ</Sup>+ln<sup>q< X+(Taylor & Francis Ltd, 2006) Tas, Kenan; Fisher, BrianLet f and g be distributions and g(n) = (g*delta(n))(x), where delta(n)(x ) is a certain sequence converging to the Dirac delta-function. The non-commutative neutrix product f o g of f and g is defined to be the neutrix limit of the sequence {fg(n) }, provided its limit h exists in the sense that [GRAPHICS] for all functions phi in D. It is proved that (x(+)(-r) ln(p) x(+)) o (x(+)(mu) ln(q) x(+)) = x(+)(-r+mu) ln(p+q) x(+) (x(-)(-r) ln(p) (x)-) o (x(-)(mu) ln(q) x(-)) = x(-)(-r+mu) ln(p+q) x(-) for mu < r - 1;mu not equal 0, +/- 1, +/- 2,..., r = 1,2,..., and p, q = 0, 1, 2,....Article Further Results on the Neutrix Composition of Distributions Involving the Delta Function and the Function Cosh+<sup>-1</Sup> (x<sup>1/R<(de Gruyter Poland Sp Z O O, 2019) Tas, Kenan; Fisher, BrianThe neutrix composition F(f(x)) of a distribution F(x) and a locally summable function f(x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {F-n(f(x))) is equal to h(x), where F-n(x) = F(x) * delta(n)(x) and {delta(n)(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function delta(x). The function cosh(+)(-1)(x + 1) is defined by cosh(+)(-1)(x+ 1) = H(x) cosh(-1)(vertical bar x vertical bar + 1), where H(x) denotes Heaviside's function. It is then proved that the neutrix composition delta((s))[cosh(+)(-1)(x(1/r) + 1)] exists and delta((s))[cosh(+)(-1)(x(1/r) + 1] = Sigma(s-1)(k=0) Sigma(kr+r-1)(j=0) Sigma(j)(i=0) (-1)(kr+r+s-j-1)r/2(j+2) ((kr + r -1)(j)) ((j)(i)) [(j - 2i + 1)(s) - (j - 2i -1)(s)]delta((k))(x) for r, s = 1, 2, .... Further results are also proved. Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh(+)(-1) (x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629-7640].Article Citation - WoS: 1Citation - Scopus: 2Some Results on the Non-Commutative Neutrix Product of Distributions(Taylor & Francis Ltd, 2009) Tas, Kenan; Fisher, BrianIt is proved that the non-commutative neutrix product of the distributions x-r and xslnq|x| exists and [image omitted] for r, q=1, 2, , s=0,1,2, and r-s1.Article Commutative Convolution of Functions and Distributions(Taylor & Francis Ltd, 2007) Tas, Kenan; Fisher, BrianThe commutative convolution f * g of two distributions f and g in D' is defined as the limit of the sequence {(f tau(n)) * (g tau(n))}, provided the limit exists, where {tau(n)} is a certain sequence of functions tn in D converging to 1. It is proved that |x|(lambda) * (sgn x|x|(-lambda-1)) = pi[cot (pi lambda) - cosec(pi lambda)] sgn x|x|(0), for lambda not equal 0, +/- 1, +/- 2, ... , where B denotes the Beta function.