On the Composition and Neutrix Composition of the Delta Function and the Function cosh^{-1}(|x|^{1/r}+1)

Main Article Content

Brian Fisher, Emin Ozcag, Fatma Al-Sirehy

Abstract

Let $F$ be a distribution in $\mathcal{D'}$ 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 $\{ F_{n}(f(x))\}$ is equal to $h(x)$, where $F_n(x) =F(x)*\delta _n(x)$ for $n=1,2, \ldots$ and $\{\delta_n(x)\}$ is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition $ \delta^{(s)}[\cosh^{-1} (x_+^{1/r}+1)] $ exists and

\beqa \delta^{(s)}[\cosh^{-1} (x_+^{1/r}+1)] = - \sum _{k=0} ^{M-1} \sum_{i=0}^{kr+r} {k \choose i}{(-1)^{i+k}rc_{r,s,k} \over (kr+r)k!}\delta ^{(k)}(x),

for $s =M-1,M, M+1,\ldots$ and $r=1,2,\ldots,$ where

$$c_{r,s,k}=\sum _{j=0}^{i} {i \choose j}{ (-1)^{kr+r-i}(2j-i)^{s+1}\over 2^{s+i+1} },$$ $M$ is the smallest integer for which $s-2r+1 < 2Mr$ and $r\le s/(2M+2).$

Further results are also proved.

Article Details

References

  1. P. Antosik, Composition of Distributions, Technical Report no.9 University of Wisconsin, Milwaukee, (1988-89), pp.1-30.
  2. P. Antosik, J. Mikusinski and R. Sikorski, Theory of Distributions, The sequential Approach, PWN-Elsevier, Warszawa-Amsterdam (1973).
  3. J.G. van der Corput, Introduction to the neutrix calculus, J. Analyse Math., 7(1959), 291-398.
  4. B. Fisher, On defining the distribution δ (r) (f(x)), Rostock. Math. Kolloq., 23(1983), 73-80.
  5. B. Fisher, On defining the change of variable in distributions, Rostock. Math. Kolloq., 28(1985), 33-40.
  6. B. Fisher, The delta function and the composition of distributions, Dem. Math., 35(1)(2002), 117-123.
  7. B. Fisher, The composition and neutrix composition of distributions, in: Kenan Ta ¸ s et al. (Eds.), Mathematical Methods in Engineering, Springer, Dordrecht, 2007, pp. 59-69.
  8. B. Fisher and B. Jolevska-Tuneska, Two results on the composition of distributions, Thai. J. Math., 3(1)(2005), 17-26.
  9. B. Fisher and A. Kilicman, On the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function, Integral Transforms Spec. Funct. 21(12)(2010), 935-944.
  10. H. Kleinert, A. Chervyakov, Rules for integrals over products of distributions from coordinate independence of path integrals, Eur. Phys. J. C Part. Fields 19(4)(2001), 743-747.
  11. B. Fisher, A. Kananthai, G. Sritanatana and K. Nonlaopon, The composition of the distributions $x_-^{ms}ln x_-$ and $x_+^{r-p/m}$, Integral Transforms Spec. Funct., 16(1)(2005), 13-20.
  12. B. Fisher and E. Ozcag, Some results on the neutrix composition of the delta function, Filomat, 26(6)(2012), 1247-1256.
  13. B. Fisher and K. Ta ¸ s, On the composition of the distributions $x^{-1}ln|x|$ and $x_+^r$, Integral Transforms Spec. Funct., 16(7)(2005), 533-543.
  14. S. Gasiorowics, Elementery Particle Physics, John Wiley and Sons, New York, 1966.
  15. I.M. Gel'fand and G.E. Shilov, Generalized Functions, Volume I, Academic Press, New York and London, 1st edition, 1964.
  16. D. S. Jones, Hadamard's Finite Part., Math. Methods Appl. Sci. 19(13)(1996), 1017-1052.
  17. E. L. Koh and C. K. Li, On distributions δ k and (δ 0 ) k , Math. Nachr., 157(1992), 243-248.
  18. H. Kou and B. Fisher, On Composition of Distributions, Publ. Math. Debrecen, 40(3-4)(1992), 279-290.
  19. L. Lazarova, B. Jolevska-Tuneska, I. Akturk and E. Ozcag Note on the Distribution Composition (x µ+) λ. Bull. Malaysian Math. Soc., (2016). doi:10.1007/s40840-016-0342-2.
  20. C. K. Li and C. Li, On defining the distributions δ k and (δ 0 ) k by fractional derivatives, Appl. Math. Compt., 246(2014), 502-513.
  21. E. Ozcag, Defining the k-th powers of the Dirac-delta distribution for negative integers, Appl. Math. Letters, 14(2001), 419-423.
  22. E. Ozcag, U. Gulen and B. Fisher, On the distribution $delta_+^k$, Integral Transforms Spec. Funct., 9(2000), 57-64.
  23. E. Ozcag, L. Lazarova and B. Jolevska-Tuneska, Defining Compositions of $x_+^mu, |x|^mu, x^{-s}$ and $x^{-s}ln |x|$ as Neutrix Limit of Regular Sequences, Commun. Math. Stat., 4(1)(2016), 63-80.
  24. G. Temple, The Theory of generalized Functions, Proc. Roy. Soc. ser. A 28(1955), 175-190.