{"id":"https://openalex.org/W4224320930","doi":"https://doi.org/10.3390/sym14050869","title":"Unified Integrals of Generalized Mittag\u2013Leffler Functions and Their Graphical Numerical Investigation","display_name":"Unified Integrals of Generalized Mittag\u2013Leffler Functions and Their Graphical Numerical Investigation","publication_year":2022,"publication_date":"2022-04-23","ids":{"openalex":"https://openalex.org/W4224320930","doi":"https://doi.org/10.3390/sym14050869"},"language":"en","primary_location":{"id":"doi:10.3390/sym14050869","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050869","pdf_url":"https://www.mdpi.com/2073-8994/14/5/869/pdf?version=1650937387","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/5/869/pdf?version=1650937387","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075128940","display_name":"Nabiullah Khan","orcid":"https://orcid.org/0000-0003-0389-7899"},"institutions":[{"id":"https://openalex.org/I171210897","display_name":"Aligarh Muslim University","ror":"https://ror.org/03kw9gc02","country_code":"IN","type":"education","lineage":["https://openalex.org/I171210897"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Nabiullah Khan","raw_affiliation_strings":["Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India","institution_ids":["https://openalex.org/I171210897"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101465288","display_name":"Mohammad Iqbal Khan","orcid":"https://orcid.org/0000-0001-5469-1976"},"institutions":[{"id":"https://openalex.org/I171210897","display_name":"Aligarh Muslim University","ror":"https://ror.org/03kw9gc02","country_code":"IN","type":"education","lineage":["https://openalex.org/I171210897"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Mohammad Iqbal Khan","raw_affiliation_strings":["Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India","institution_ids":["https://openalex.org/I171210897"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082976722","display_name":"Talha Usman","orcid":"https://orcid.org/0000-0002-4208-6784"},"institutions":[{"id":"https://openalex.org/I4403386702","display_name":"University of Technology and Applied Sciences-Suhar","ror":"https://ror.org/018g8cj68","country_code":"OM","type":"education","lineage":["https://openalex.org/I4403386702","https://openalex.org/I4405258755"]}],"countries":["OM"],"is_corresponding":false,"raw_author_name":"Talha Usman","raw_affiliation_strings":["Department of General Requirements, University of Technology and Applied Sciences-Sur, Muscat 133, Oman"],"raw_orcid":"https://orcid.org/0000-0002-4208-6784","affiliations":[{"raw_affiliation_string":"Department of General Requirements, University of Technology and Applied Sciences-Sur, Muscat 133, Oman","institution_ids":["https://openalex.org/I4403386702"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089496116","display_name":"Kamsing Nonlaopon","orcid":"https://orcid.org/0000-0002-7469-5402"},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Kamsing Nonlaopon","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":"https://orcid.org/0000-0002-7469-5402","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042854866","display_name":"Shrideh Al\u2010Omari","orcid":"https://orcid.org/0000-0001-8955-5552"},"institutions":[{"id":"https://openalex.org/I33926330","display_name":"Al-Balqa Applied University","ror":"https://ror.org/00qedmt22","country_code":"JO","type":"education","lineage":["https://openalex.org/I33926330"]}],"countries":["JO"],"is_corresponding":false,"raw_author_name":"Shrideh Al-Omari","raw_affiliation_strings":["Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan"],"raw_orcid":"https://orcid.org/0000-0001-8955-5552","affiliations":[{"raw_affiliation_string":"Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan","institution_ids":["https://openalex.org/I33926330"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5089496116"],"corresponding_institution_ids":["https://openalex.org/I179193067"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.406,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.52404944,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"14","issue":"5","first_page":"869","last_page":"869"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9987000226974487,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6644807457923889},{"id":"https://openalex.org/keywords/gaussian-integral","display_name":"Gaussian integral","score":0.6360548734664917},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5518671274185181},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.5443503260612488},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.4860828220844269},{"id":"https://openalex.org/keywords/order-of-integration","display_name":"Order of integration (calculus)","score":0.47980716824531555},{"id":"https://openalex.org/keywords/trigonometric-integral","display_name":"Trigonometric integral","score":0.47040683031082153},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.4457830488681793},{"id":"https://openalex.org/keywords/volume-integral","display_name":"Volume integral","score":0.43762633204460144},{"id":"https://openalex.org/keywords/multiple-integral","display_name":"Multiple integral","score":0.4306657016277313},{"id":"https://openalex.org/keywords/gaussian-quadrature","display_name":"Gaussian quadrature","score":0.42775222659111023},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.41731277108192444},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3752920925617218},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.33295759558677673},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.2616983652114868},{"id":"https://openalex.org/keywords/nystr\u00f6m-method","display_name":"Nystr\u00f6m method","score":0.1355864405632019},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0699593722820282}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6644807457923889},{"id":"https://openalex.org/C70139506","wikidata":"https://www.wikidata.org/wiki/Q1060321","display_name":"Gaussian integral","level":2,"score":0.6360548734664917},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5518671274185181},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.5443503260612488},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4860828220844269},{"id":"https://openalex.org/C38582570","wikidata":"https://www.wikidata.org/wiki/Q3825182","display_name":"Order of integration (calculus)","level":2,"score":0.47980716824531555},{"id":"https://openalex.org/C65692021","wikidata":"https://www.wikidata.org/wiki/Q1094282","display_name":"Trigonometric integral","level":3,"score":0.47040683031082153},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.4457830488681793},{"id":"https://openalex.org/C51577431","wikidata":"https://www.wikidata.org/wiki/Q1788036","display_name":"Volume integral","level":3,"score":0.43762633204460144},{"id":"https://openalex.org/C54865166","wikidata":"https://www.wikidata.org/wiki/Q844937","display_name":"Multiple integral","level":2,"score":0.4306657016277313},{"id":"https://openalex.org/C167196314","wikidata":"https://www.wikidata.org/wiki/Q767680","display_name":"Gaussian quadrature","level":4,"score":0.42775222659111023},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41731277108192444},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3752920925617218},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.33295759558677673},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.2616983652114868},{"id":"https://openalex.org/C48265008","wikidata":"https://www.wikidata.org/wiki/Q7071295","display_name":"Nystr\u00f6m method","level":3,"score":0.1355864405632019},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0699593722820282},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14050869","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050869","pdf_url":"https://www.mdpi.com/2073-8994/14/5/869/pdf?version=1650937387","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:234c086d3b48425b841246e47e9e8e02","is_oa":true,"landing_page_url":"https://doaj.org/article/234c086d3b48425b841246e47e9e8e02","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":"Symmetry, Vol 14, Iss 5, p 869 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/5/869/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14050869","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14050869","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14050869","pdf_url":"https://www.mdpi.com/2073-8994/14/5/869/pdf?version=1650937387","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4224320930.pdf"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W69681244","https://openalex.org/W605875166","https://openalex.org/W1980023378","https://openalex.org/W1993621494","https://openalex.org/W2016585644","https://openalex.org/W2041454331","https://openalex.org/W2045350842","https://openalex.org/W2076453374","https://openalex.org/W2091714372","https://openalex.org/W2131443853","https://openalex.org/W2131927298","https://openalex.org/W2170541036","https://openalex.org/W2579796401","https://openalex.org/W2609141206","https://openalex.org/W2898288362","https://openalex.org/W2917167622","https://openalex.org/W2945166078","https://openalex.org/W2953380742","https://openalex.org/W2964817671","https://openalex.org/W3008053174","https://openalex.org/W3013678354","https://openalex.org/W3087456302","https://openalex.org/W3092272797","https://openalex.org/W3110149628","https://openalex.org/W3185010451","https://openalex.org/W3196460261","https://openalex.org/W3205957286","https://openalex.org/W4200250901","https://openalex.org/W6602848315","https://openalex.org/W6683520554","https://openalex.org/W7000791799","https://openalex.org/W7058483885"],"related_works":["https://openalex.org/W4386782986","https://openalex.org/W3128814429","https://openalex.org/W560143338","https://openalex.org/W3128097264","https://openalex.org/W2992193318","https://openalex.org/W2606175160","https://openalex.org/W2569355762","https://openalex.org/W2012994876","https://openalex.org/W3004420723","https://openalex.org/W2071825966"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3,41],"obtain":[4],"certain":[5],"finite":[6],"integrals":[7,23,45,58],"concerning":[8],"generalized":[9,19],"Mittag\u2013Leffler":[10],"functions,":[11],"which":[12],"are":[13,26],"evaluated":[14],"in":[15,28],"terms":[16],"of":[17,24],"the":[18,47,56],"Fox\u2013Wright":[20],"function.":[21],"The":[22],"concern":[25],"unified":[27],"nature":[29],"and":[30,51],"thereby":[31],"yield":[32],"some":[33,44],"new":[34],"integral":[35],"formulas":[36],"as":[37],"special":[38],"cases.":[39],"Moreover,":[40],"numerically":[42],"compute":[43],"using":[46,60],"Gaussian":[48],"quadrature":[49],"formula":[50],"draw":[52],"a":[53],"comparison":[54],"with":[55],"main":[57],"by":[59],"graphical":[61],"numerical":[62],"investigation.":[63]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2022-04-26T00:00:00"}