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Title: | On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions |

Authors: | Fisher, Brian Kılıcman, Adem |

First Published: | 2011 |

Publisher: | Hindawi Publishing Corporation |

Citation: | Journal of Applied Mathematics, Volume 2011, Article ID 612353 |

Abstract: | Let F be a distribution in D[superscript 1] 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[subscript n](f(x))} is equal to h(x), where F[subscript n](x)= F(x) ∗ δ[subscript n](x) for n = 1, 2, . . . and {δ[subscript n](x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] does not exists. In this study, it is proved that the neutrix composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] exists and is given by δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] = ∑[superscript sr+r-1, subscript k=0] ∑[superscript k, subscript i=0] ([superscript k, subscript i]) ((-1)[superscript k] rc[subscript s,k,i]/2[superscript k+1]k!)δ([superscript k])(x), for s = 0, 1, 2, . . . and r = 1, 2, . . ., where c[subscript s,k,i] = (−1)[superscript s]s![(k − 2i + 1)[superscript rs−1] + (k − 2i − 1)[superscript rs+r−1]/(2(rs + r − 1)!). Further results are also proved. |

DOI Link: | 10.1155/2011/612353 |

ISSN: | 1110-757X |

eISSN: | 1687-0042 |

Links: | http://www.hindawi.com/journals/jam/2011/612353/ http://hdl.handle.net/2381/28476 |

Version: | Publisher Version |

Status: | Peer-reviewed |

Type: | Journal Article |

Rights: | Copyright © 2011 B. Fisher and A. Kılıcman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |

Appears in Collections: | Published Articles, Dept. of Mathematics |

Files in This Item:

File | Description | Size | Format | |
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10.1155_2011_612353.pdf | Published (publisher PDF) | 1.29 MB | Adobe PDF | View/Open |

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