{"id":"https://openalex.org/W4285325472","doi":"https://doi.org/10.2478/forma-2021-0019","title":"Improper Integral. Part I","display_name":"Improper Integral. Part I","publication_year":2021,"publication_date":"2021-12-01","ids":{"openalex":"https://openalex.org/W4285325472","doi":"https://doi.org/10.2478/forma-2021-0019"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2021-0019","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0019","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0019","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0019","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015450295","display_name":"Noboru Endou","orcid":"https://orcid.org/0000-0002-5922-2332"},"institutions":[{"id":"https://openalex.org/I4210113483","display_name":"National Institute of Technology, Gifu College","ror":"https://ror.org/025b2hx26","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210113483","https://openalex.org/I4210120810"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Noboru Endou","raw_affiliation_strings":["National Institute of Technology, Gifu College , 2236-2 Kamimakuwa, Motosu , Gifu , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"National Institute of Technology, Gifu College , 2236-2 Kamimakuwa, Motosu , Gifu , Japan","institution_ids":["https://openalex.org/I4210113483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5015450295"],"corresponding_institution_ids":["https://openalex.org/I4210113483"],"apc_list":null,"apc_paid":null,"fwci":0.5985,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.69317195,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"29","issue":"4","first_page":"201","last_page":"220"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9918000102043152,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11424","display_name":"Security and Verification in Computing","score":0.9901000261306763,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/mathematics","display_name":"Mathematics","score":0.8419290781021118},{"id":"https://openalex.org/keywords/riemann-integral","display_name":"Riemann integral","score":0.751268744468689},{"id":"https://openalex.org/keywords/improper-integral","display_name":"Improper integral","score":0.7283246517181396},{"id":"https://openalex.org/keywords/volume-integral","display_name":"Volume integral","score":0.715255618095398},{"id":"https://openalex.org/keywords/multiple-integral","display_name":"Multiple integral","score":0.6476145386695862},{"id":"https://openalex.org/keywords/daniell-integral","display_name":"Daniell integral","score":0.6031103134155273},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.5770480632781982},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.5331529974937439},{"id":"https://openalex.org/keywords/line-integral","display_name":"Line integral","score":0.5182897448539734},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.4560146629810333},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.455051988363266},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42511850595474243},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.40613630414009094},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.402154803276062},{"id":"https://openalex.org/keywords/singular-integral","display_name":"Singular integral","score":0.1873316466808319},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.07115927338600159}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8419290781021118},{"id":"https://openalex.org/C41833934","wikidata":"https://www.wikidata.org/wiki/Q697181","display_name":"Riemann integral","level":4,"score":0.751268744468689},{"id":"https://openalex.org/C142017724","wikidata":"https://www.wikidata.org/wiki/Q464118","display_name":"Improper integral","level":4,"score":0.7283246517181396},{"id":"https://openalex.org/C51577431","wikidata":"https://www.wikidata.org/wiki/Q1788036","display_name":"Volume integral","level":3,"score":0.715255618095398},{"id":"https://openalex.org/C54865166","wikidata":"https://www.wikidata.org/wiki/Q844937","display_name":"Multiple integral","level":2,"score":0.6476145386695862},{"id":"https://openalex.org/C134810832","wikidata":"https://www.wikidata.org/wiki/Q2996956","display_name":"Daniell integral","level":4,"score":0.6031103134155273},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5770480632781982},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.5331529974937439},{"id":"https://openalex.org/C75023562","wikidata":"https://www.wikidata.org/wiki/Q467699","display_name":"Line integral","level":3,"score":0.5182897448539734},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4560146629810333},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.455051988363266},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42511850595474243},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.40613630414009094},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.402154803276062},{"id":"https://openalex.org/C92571104","wikidata":"https://www.wikidata.org/wiki/Q888122","display_name":"Singular integral","level":3,"score":0.1873316466808319},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.07115927338600159},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2021-0019","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0019","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0019","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/13660","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/13660","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"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":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2021-0019","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0019","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0019","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.4000000059604645,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4285325472.pdf","grobid_xml":"https://content.openalex.org/works/W4285325472.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2125669297","https://openalex.org/W2146235261","https://openalex.org/W2295728669","https://openalex.org/W2770430340"],"related_works":["https://openalex.org/W2393558284","https://openalex.org/W2357078175","https://openalex.org/W2382402128","https://openalex.org/W1970788826","https://openalex.org/W1552473859","https://openalex.org/W2362668352","https://openalex.org/W2047416813","https://openalex.org/W560143338","https://openalex.org/W2211673120","https://openalex.org/W3177264043"],"abstract_inverted_index":{"Summary":[0],"In":[1,60],"this":[2],"article,":[3],"we":[4,43,89],"deal":[5],"with":[6,19],"Riemann\u2019s":[7],"improper":[8,33,47,79],"integral":[9,54,80],"[1],":[10],"using":[11],"the":[12,53,68,78,86],"Mizar":[13],"system":[14],"[2],":[15],"[3].":[16],"Improper":[17],"integrals":[18,34,48],"finite":[20,58],"values":[21],"are":[22,40],"discussed":[23],"in":[24,31,64],"[5]":[25,65],"by":[26],"Yamazaki":[27],"et":[28],"al.,":[29],"but":[30,76],"general,":[32],"do":[35],"not":[36,51],"assume":[37],"that":[38,49,67],"they":[39],"finite.":[41],"Therefore,":[42],"have":[44],"formalized":[45],"general":[46],"does":[50],"limit":[52],"value":[55],"to":[56],"a":[57,73],"value.":[59],"addition,":[61],"each":[62],"theorem":[63],"assumes":[66],"domain":[69],"of":[70],"integrand":[71],"includes":[72],"closed":[74],"interval,":[75,88],"since":[77],"should":[81],"be":[82],"discusses":[83],"based":[84],"on":[85],"half-open":[87],"also":[90],"corrected":[91],"it.":[92]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}