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Title: | On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions |

Authors: | Fisher, Brian Kılıcman, Adem |

First Published: | 2011 |

Publisher: | Hindawi Publishing Corporation |

Citation: | Journal of Applied Mathematics, Volume 2011, Article ID 846736 |

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. It is proved that the neutrix composition δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) exists and δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) = ∑[superscript s-1, subscript k=0] ∑[superscript K[subscript k], subscript i=0] ((-1)[superscript k]c[subscript s-2i-1,k] (rs)!/2sk!)δ([superscript k])(x) for r,s = 1, 2, . . ., where K[subscript k] is the integer part of (s-k-1)/2 and the constants c[subscript j,k] are defined by the expansion (tanh[superscript -1]x)superscript k = {∑[superscript ∞, subscript i=0] (x[superscript 2i+1]/(2i + 1))}[superscript k] = ∑[superscript ∞, subscript j=k] c[subscript j,k]x[superscript j], for k = 0,1,2,.... Further results also provided. |

DOI Link: | 10.1155/2011/846736 |

ISSN: | 1110-757X |

eISSN: | 1687-0042 |

Links: | http://www.hindawi.com/journals/jam/2011/846736/ http://hdl.handle.net/2381/28477 |

Version: | Publisher Version |

Status: | Peer-reviewed |

Type: | Journal Article |

Rights: | Copyright © 2011 B. Fisher and A. Kılıc¸man. 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_846736.pdf | Published (publisher PDF) | 1.29 MB | Adobe PDF | View/Open |

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