Brian Fisheraand Biljana JOLEVSKATUNESKAB

RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION

Anahtar Kelimeler:

-

RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION

The neutrix composition F (f (x))) of a distribution F (x) and a locallysummable function f (x) is said to exist and be equal to the distributionh(x) if the neutrix limit of the sequence {Fn (f (x))} is equal to h(x),where Fn (x) = F (x) ∗ δn (x) and {δn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). It is proved that the neutrix composition δ(s) {[exp + (x) − 1]r } exists andδ (s) {[exp + (x) − 1]r } = rs+r−1k=0 (−1)s+k s!c rs+r−1,k2rk! δ (k) (x), for r = 1, 2, . . . and s = 0, 1, 2, . . .. Further results are also proved.

Keywords:

distribution, dirac-delta function, composition of distributions, neutrix neutrix limit.,

PDF

___

van der Corput, J. G. Introduction to the neutrix calculus, J. Analyse Math., 7, 291–398, 195 Fisher, B. On defining the change of variable in distributions, Rostock. Math. Kolloq., 28, 75–86, 1985.
Fisher, B. On defining the distribution δ (r) (f (x)), Rostock. Math. Kolloq., 23, 73–80, 1993. Fisher, B. On defining the distribution (x r + ) −s − , Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 15, 119–129, 1985.
Fisher, B. The delta function and the composition of distributions, Dem. Math. 35(1), 117– 123, 2002.
Fisher, B. The composition and neutrix composition of distributions, Proc. Math. Methods Engineering, Springer, 59–69, 2007.
Fisher B. and Kılı¸ cman, A. On the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function, Integral Transforms Spec. Funct., 21(12), 935–944, 2010.
Fisher, B., Kraiweeradechachai, T., and ¨ Oz¸ ca¯ g, E. Results on the neutrix composition of the delta function, Hacet. J. Math. Stat., 36(2), 147–156, 2007.
Fisher, B., Jolevska-Tuneska, B., and ¨ Oz¸ ca¯ g, E. Further results on the composition of distributions, Integral Transforms and special functions, 13 (2), 109–116, 2002
Gel’fand, I. M. and Shilov, G. E. Generalized Functions, Vol. I, (Academic Press, 1964).
Jolevska-Tuneska, B. and Ozcag, E. On the composition of distributions x −s ln |x| and |x| µ , International Journal of Mathematics and Mathematical Sciences, 2007, 9 pages., doi:1155/2007/60129
Kraiweeradechachai, T., Orankitjaroen, S., Fisher. B., and ¨ Oz¸ ca¯ g, E. Further results on the neutrix composition of the delta function, East-West J. Math., 11(2), 151–164, 2009.
Ng, Jack Y. and van Dam, H. Neutrix calculus and finite quantum field theory, J. Phys. A: Math. Gen. 38, 317–323, 2005.
Ng, Jack Y. and van Dam, H. An application of neutrix calculus to quantum field theory, International Journal of Modern Physics A, 21 (2), 297–312, 2006.