{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:51:45Z","timestamp":1753883505228,"version":"3.41.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,6,1]],"date-time":"2011-06-01T00:00:00Z","timestamp":1306886400000},"content-version":"vor","delay-in-days":151,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004530","name":"Universiti Putra Malaysia","doi-asserted-by":"publisher","award":["05-01-09-0720RU"],"award-info":[{"award-number":["05-01-09-0720RU"]}],"id":[{"id":"10.13039\/501100004530","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>Let <jats:italic>F<\/jats:italic> be a distribution in <jats:italic>\ud835\udc9f<\/jats:italic><jats:italic>\u2032<\/jats:italic> and let <jats:italic>f<\/jats:italic> be a locally summable function. The composition <jats:italic>F<\/jats:italic>(<jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>)) of <jats:italic>F<\/jats:italic> and <jats:italic>f<\/jats:italic> is said to exist and be equal to the distribution <jats:italic>h<\/jats:italic>(<jats:italic>x<\/jats:italic>) if the limit of the sequence {<jats:italic>F<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>))} is equal to <jats:italic>h<\/jats:italic>(<jats:italic>x<\/jats:italic>), where <jats:italic>F<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) = <jats:italic>F<\/jats:italic>(<jats:italic>x<\/jats:italic>)*<jats:italic>\u03b4<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) for <jats:italic>n<\/jats:italic> = 1,2, \u2026 and {<jats:italic>\u03b4<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition <jats:italic>\u03b4<\/jats:italic><jats:sup>(<jats:italic>s<\/jats:italic>)<\/jats:sup>[(sinh\u2009<jats:sup>\u22121<\/jats:sup><jats:italic>x<\/jats:italic><jats:sub>+<\/jats:sub>)\u2009<jats:sup><jats:italic>r<\/jats:italic><\/jats:sup>] does not exists. In this study, it is proved that the neutrix composition <jats:italic>\u03b4<\/jats:italic><jats:sup>(<jats:italic>s<\/jats:italic>)<\/jats:sup>[(sinh\u2009<jats:sup>\u22121<\/jats:sup><jats:italic>x<\/jats:italic><jats:sub>+<\/jats:sub>)\u2009<jats:sup><jats:italic>r<\/jats:italic><\/jats:sup>] exists and is given by , for <jats:italic>s<\/jats:italic> = 0,1, 2, \u2026 and <jats:italic>r<\/jats:italic> = 1,2, \u2026, where <jats:italic>c<\/jats:italic><jats:sub><jats:italic>s<\/jats:italic>,<jats:italic>k<\/jats:italic>,<jats:italic>i<\/jats:italic><\/jats:sub> = (\u22121)\u2009<jats:sup><jats:italic>s<\/jats:italic><\/jats:sup><jats:italic>s<\/jats:italic>! [(<jats:italic>k<\/jats:italic> \u2212 2<jats:italic>i<\/jats:italic> + 1)\u2009<jats:sup><jats:italic>r<\/jats:italic><jats:italic>s<\/jats:italic>\u22121<\/jats:sup> + (<jats:italic>k<\/jats:italic> \u2212 2<jats:italic>i<\/jats:italic> \u2212 1)\u2009<jats:sup><jats:italic>r<\/jats:italic><jats:italic>s<\/jats:italic>+<jats:italic>r<\/jats:italic>\u22121<\/jats:sup>]\/(2(<jats:italic>r<\/jats:italic><jats:italic>s<\/jats:italic> + <jats:italic>r<\/jats:italic> \u2212 1)!). Further results are also proved.<\/jats:p>","DOI":"10.1155\/2011\/612353","type":"journal-article","created":{"date-parts":[[2011,6,1]],"date-time":"2011-06-01T19:33:42Z","timestamp":1306956822000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions"],"prefix":"10.1155","volume":"2011","author":[{"given":"Brian","family":"Fisher","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adem","family":"K\u0131l\u0131\u00e7man","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,6]]},"reference":[{"key":"e_1_2_4_1_2","doi-asserted-by":"publisher","DOI":"10.1080\/10652460903335061"},{"key":"e_1_2_4_2_2","first-page":"753","article-title":"On the commutative neutrix product of distributions","volume":"30","author":"K\u0131l\u0131\u00e7man A.","year":"1999","journal-title":"Indian Journal of Pure and Applied Mathematics"},{"key":"e_1_2_4_3_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013719619356"},{"key":"e_1_2_4_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2008.05.048"},{"key":"e_1_2_4_5_2","unstructured":"AntosikP. 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A.","family":"Soliman","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1155\/2011\/612353"],"URL":"https:\/\/doi.org\/10.1155\/2011\/612353","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2011,1]]},"assertion":[{"value":"2010-12-24","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-04-08","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-06-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"612353"}}