{"id":"https://openalex.org/W2151969911","doi":"https://doi.org/10.1155/2011/846736","title":"On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions","display_name":"On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions","publication_year":2011,"publication_date":"2011-01-01","ids":{"openalex":"https://openalex.org/W2151969911","doi":"https://doi.org/10.1155/2011/846736","mag":"2151969911"},"language":"en","primary_location":{"id":"doi:10.1155/2011/846736","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2011/846736","pdf_url":"https://downloads.hindawi.com/journals/jam/2011/846736.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2011/846736.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072243347","display_name":"Brian Fisher","orcid":"https://orcid.org/0000-0001-5665-5709"},"institutions":[{"id":"https://openalex.org/I153648349","display_name":"University of Leicester","ror":"https://ror.org/04h699437","country_code":"GB","type":"education","lineage":["https://openalex.org/I153648349"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Brian Fisher","raw_affiliation_strings":["Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK","Department of Mathematics, University of Leicester, Leicester LE1 7RH"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK","institution_ids":["https://openalex.org/I153648349"]},{"raw_affiliation_string":"Department of Mathematics, University of Leicester, Leicester LE1 7RH","institution_ids":["https://openalex.org/I153648349"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025569523","display_name":"Adem K\u0131l\u0131\u00e7man","orcid":"https://orcid.org/0000-0002-1217-963X"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Adem K\u0131l\u0131\u00e7man","raw_affiliation_strings":["Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400 Selangor"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400 Selangor","institution_ids":["https://openalex.org/I130343225"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5025569523"],"corresponding_institution_ids":["https://openalex.org/I130343225"],"apc_list":{"value":1025,"currency":"USD","value_usd":1025},"apc_paid":{"value":1025,"currency":"USD","value_usd":1025},"fwci":0.4087,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.60280688,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"2011","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.9624000191688538,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9157000184059143,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5903560519218445},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3347090184688568},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2683216333389282}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5903560519218445},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3347090184688568},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2683216333389282}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1155/2011/846736","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2011/846736","pdf_url":"https://downloads.hindawi.com/journals/jam/2011/846736.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:figshare.com:article/10158359","is_oa":true,"landing_page_url":null,"pdf_url":"https://figshare.com/articles/journal_contribution/On_the_composition_and_neutrix_composition_of_the_delta_function_with_the_hyperbolic_tangent_and_its_inverse_functions/10158359","source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:CULeuclid:euclid.jam/1331818616","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jam/1331818616","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:846736","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2011/846736.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:12d3253e2b0e4272b100178e3f86b06b","is_oa":true,"landing_page_url":"https://doaj.org/article/12d3253e2b0e4272b100178e3f86b06b","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Applied Mathematics, Vol 2011 (2011)","raw_type":"article"},{"id":"pmh:oai:lra.le.ac.uk:2381/28477","is_oa":true,"landing_page_url":"http://www.hindawi.com/journals/jam/2011/846736/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402365","display_name":"Leicester Research Archive (University of Leicester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153648349","host_organization_name":"University of Leicester","host_organization_lineage":["https://openalex.org/I153648349"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal Article"}],"best_oa_location":{"id":"doi:10.1155/2011/846736","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2011/846736","pdf_url":"https://downloads.hindawi.com/journals/jam/2011/846736.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3432821215","display_name":null,"funder_award_id":"05-01-09-0720RU","funder_id":"https://openalex.org/F4320322714","funder_display_name":"Universiti Putra Malaysia"}],"funders":[{"id":"https://openalex.org/F4320322714","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2151969911.pdf","grobid_xml":"https://content.openalex.org/works/W2151969911.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W10654363","https://openalex.org/W282947517","https://openalex.org/W1485851255","https://openalex.org/W1966095617","https://openalex.org/W2000124520","https://openalex.org/W2012386126","https://openalex.org/W2012407322","https://openalex.org/W2018890580","https://openalex.org/W2027496368","https://openalex.org/W2027631244","https://openalex.org/W2044441912","https://openalex.org/W2047679447","https://openalex.org/W2088159024","https://openalex.org/W2118597446","https://openalex.org/W2152024720","https://openalex.org/W2170735729","https://openalex.org/W2613064678","https://openalex.org/W4241060531","https://openalex.org/W4406562552","https://openalex.org/W6685091601","https://openalex.org/W7046486563"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2544423928","https://openalex.org/W2052122378","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W4313731559"],"abstract_inverted_index":{"Let":[0],"F":[1,18,25,49,64,70],"be":[2,11,33],"a":[3,12,92],"distribution":[4,37],"in":[5],"\ud835\udc9f":[6],"\u2032":[7],"and":[8,26,32,84,122,144],"let":[9],"f":[10,20,27,52],"locally":[13],"summable":[14],"function.":[15,101],"The":[16],"composition":[17,108],"(":[19,21,39,51,53,60,66,71,76,88,110,138],"x":[22,40,54,61,67,72,77,89,115],"))":[23],"of":[24,45,137],"is":[28,56,91,103,133],"said":[29],"to":[30,35,58,97],"exist":[31],"equal":[34,57],"the":[36,43,46,98,106,134,145,154],"h":[38,59],")":[41,68,78,117,120],"if":[42],"limit":[44],"sequence":[47,95],"{":[48,85],"n":[50,65,75,80,87],"))}":[55],"),":[62],"where":[63,130],"=":[69,81,127,159],")*":[73],"\u03b4":[74,86,109],"for":[79,123,157],"1,2,":[82,128],"\u2026":[83],")}":[90],"certain":[93],"regular":[94],"converging":[96],"Dirac":[99],"delta":[100],"It":[102],"proved":[104],"that":[105],"neutrix":[107],"r":[111,119,124],"s":[112,126,139],"\u22121)":[113],"((tanh":[114],"+":[116],"1/":[118],"exists":[121],",":[125,149,156],"\u2026,":[129],"K":[131],"k":[132,141,150,158],"integer":[135],"part":[136],"\u2212":[140,142],"1)/2":[143],"constants":[146],"c":[147],"j":[148],"are":[151,165],"defined":[152],"by":[153],"expansion":[155],"0,1,":[160],"2,":[161],"\u2026.":[162],"Further":[163],"results":[164],"also":[166],"proved.":[167]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-03T08:13:44.112507","created_date":"2025-10-10T00:00:00"}
