{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:55:59Z","timestamp":1753883759021,"version":"3.41.2"},"reference-count":22,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,8,1]],"date-time":"2011-08-01T00:00:00Z","timestamp":1312156800000},"content-version":"vor","delay-in-days":212,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004530","name":"University Putra Malaysia","doi-asserted-by":"crossref","award":["05-01-09-0720RU"],"award-info":[{"award-number":["05-01-09-0720RU"]}],"id":[{"id":"10.13039\/501100004530","id-type":"DOI","asserted-by":"crossref"}]}],"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. It is proved that the neutrix composition <jats:italic>\u03b4<\/jats:italic><jats:sup>(<jats:italic>r<\/jats:italic><jats:italic>s<\/jats:italic>\u22121)<\/jats:sup>((tanh<jats:italic>x<\/jats:italic><jats:sub>+<\/jats:sub>)\u2009<jats:sup>1\/<jats:italic>r<\/jats:italic><\/jats:sup>) exists and  for <jats:italic>r<\/jats:italic>, <jats:italic>s<\/jats:italic> = 1,2, \u2026, where <jats:italic>K<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> is the integer part of (<jats:italic>s<\/jats:italic> \u2212 <jats:italic>k<\/jats:italic> \u2212 1)\/2 and the constants <jats:italic>c<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic>,<jats:italic>k<\/jats:italic><\/jats:sub> are defined by the expansion , for <jats:italic>k<\/jats:italic> = 0,1, 2, \u2026. Further results are also proved.<\/jats:p>","DOI":"10.1155\/2011\/846736","type":"journal-article","created":{"date-parts":[[2011,8,1]],"date-time":"2011-08-01T19:01:04Z","timestamp":1312225264000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse 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":[[2011,8]]},"reference":[{"key":"e_1_2_4_1_2","doi-asserted-by":"publisher","DOI":"10.1080\/10652460903335061"},{"key":"e_1_2_4_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/10652469608819095"},{"key":"e_1_2_4_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/10652469.2010.497271"},{"key":"e_1_2_4_4_2","doi-asserted-by":"publisher","DOI":"10.1155\/2011\/612353"},{"key":"e_1_2_4_5_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":"The Indian Journal of Pure and Applied Mathematics"},{"key":"e_1_2_4_6_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013719619356"},{"key":"e_1_2_4_7_2","doi-asserted-by":"publisher","DOI":"10.1080\/10652469808819202"},{"key":"e_1_2_4_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2008.05.048"},{"key":"e_1_2_4_9_2","unstructured":"AntosikP. 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