{"id":"https://openalex.org/W4402704002","doi":"https://doi.org/10.3390/axioms13090631","title":"On Closed Forms of Some Trigonometric Series","display_name":"On Closed Forms of Some Trigonometric Series","publication_year":2024,"publication_date":"2024-09-14","ids":{"openalex":"https://openalex.org/W4402704002","doi":"https://doi.org/10.3390/axioms13090631"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13090631","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090631","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/axioms13090631","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008693320","display_name":"Slobodan B. Tri\u010dkovi\u0107","orcid":"https://orcid.org/0000-0002-3766-8579"},"institutions":[{"id":"https://openalex.org/I152518017","display_name":"University of Nis","ror":"https://ror.org/00965bg92","country_code":"RS","type":"education","lineage":["https://openalex.org/I152518017"]}],"countries":["RS"],"is_corresponding":true,"raw_author_name":"Slobodan B. Tri\u010dkovi\u0107","raw_affiliation_strings":["Department of Mathematics, University of Ni\u0161, 18000 Ni\u0161, Serbia"],"raw_orcid":"https://orcid.org/0000-0002-3766-8579","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Ni\u0161, 18000 Ni\u0161, Serbia","institution_ids":["https://openalex.org/I152518017"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103461410","display_name":"Miomir S. Stankovi\u0107","orcid":null},"institutions":[{"id":"https://openalex.org/I1338511612","display_name":"Serbian Academy of Sciences and Arts","ror":"https://ror.org/05m1y4204","country_code":"RS","type":"government","lineage":["https://openalex.org/I1338511612"]}],"countries":["RS"],"is_corresponding":false,"raw_author_name":"Miomir S. Stankovi\u0107","raw_affiliation_strings":["Mathematical Institute of Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Institute of Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia","institution_ids":["https://openalex.org/I1338511612"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5008693320"],"corresponding_institution_ids":["https://openalex.org/I152518017"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.2285,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.78386005,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"13","issue":"9","first_page":"631","last_page":"631"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T14500","display_name":"Advanced Mathematical Theories","score":0.998199999332428,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/trigonometric-series","display_name":"Trigonometric series","score":0.7934314012527466},{"id":"https://openalex.org/keywords/trigonometric-functions","display_name":"Trigonometric functions","score":0.7916078567504883},{"id":"https://openalex.org/keywords/sine","display_name":"Sine","score":0.7683377265930176},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7262053489685059},{"id":"https://openalex.org/keywords/trigonometric-integral","display_name":"Trigonometric integral","score":0.7013087272644043},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.6873291730880737},{"id":"https://openalex.org/keywords/trigonometric-substitution","display_name":"Trigonometric substitution","score":0.6839618682861328},{"id":"https://openalex.org/keywords/differentiation-of-trigonometric-functions","display_name":"Differentiation of trigonometric functions","score":0.6610116362571716},{"id":"https://openalex.org/keywords/inverse-trigonometric-functions","display_name":"Inverse trigonometric functions","score":0.6425687074661255},{"id":"https://openalex.org/keywords/pythagorean-trigonometric-identity","display_name":"Pythagorean trigonometric identity","score":0.6216306090354919},{"id":"https://openalex.org/keywords/trigonometric-polynomial","display_name":"Trigonometric polynomial","score":0.5303552746772766},{"id":"https://openalex.org/keywords/trigonometry","display_name":"Trigonometry","score":0.5023410320281982},{"id":"https://openalex.org/keywords/proofs-of-trigonometric-identities","display_name":"Proofs of trigonometric identities","score":0.47810763120651245},{"id":"https://openalex.org/keywords/riemann-zeta-function","display_name":"Riemann zeta function","score":0.45177266001701355},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4205020070075989},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4156322479248047},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3991287350654602},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.11846727132797241},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0755070149898529}],"concepts":[{"id":"https://openalex.org/C120362076","wikidata":"https://www.wikidata.org/wiki/Q1149336","display_name":"Trigonometric series","level":2,"score":0.7934314012527466},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.7916078567504883},{"id":"https://openalex.org/C186661526","wikidata":"https://www.wikidata.org/wiki/Q13647261","display_name":"Sine","level":2,"score":0.7683377265930176},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7262053489685059},{"id":"https://openalex.org/C65692021","wikidata":"https://www.wikidata.org/wiki/Q1094282","display_name":"Trigonometric integral","level":3,"score":0.7013087272644043},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.6873291730880737},{"id":"https://openalex.org/C9317044","wikidata":"https://www.wikidata.org/wiki/Q1191963","display_name":"Trigonometric substitution","level":5,"score":0.6839618682861328},{"id":"https://openalex.org/C30786353","wikidata":"https://www.wikidata.org/wiki/Q3296857","display_name":"Differentiation of trigonometric functions","level":5,"score":0.6610116362571716},{"id":"https://openalex.org/C68827188","wikidata":"https://www.wikidata.org/wiki/Q674533","display_name":"Inverse trigonometric functions","level":2,"score":0.6425687074661255},{"id":"https://openalex.org/C89507098","wikidata":"https://www.wikidata.org/wiki/Q2039117","display_name":"Pythagorean trigonometric identity","level":5,"score":0.6216306090354919},{"id":"https://openalex.org/C156760230","wikidata":"https://www.wikidata.org/wiki/Q1969650","display_name":"Trigonometric polynomial","level":3,"score":0.5303552746772766},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.5023410320281982},{"id":"https://openalex.org/C164897557","wikidata":"https://www.wikidata.org/wiki/Q7250037","display_name":"Proofs of trigonometric identities","level":5,"score":0.47810763120651245},{"id":"https://openalex.org/C35235930","wikidata":"https://www.wikidata.org/wiki/Q187235","display_name":"Riemann zeta function","level":2,"score":0.45177266001701355},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4205020070075989},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4156322479248047},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3991287350654602},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.11846727132797241},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0755070149898529},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C171836373","wikidata":"https://www.wikidata.org/wiki/Q2266329","display_name":"Linear interpolation","level":3,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C49608258","wikidata":"https://www.wikidata.org/wiki/Q611705","display_name":"Bicubic interpolation","level":4,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms13090631","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090631","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:4fb7e4fb30a145c48de93e3872deb3d8","is_oa":true,"landing_page_url":"https://doaj.org/article/4fb7e4fb30a145c48de93e3872deb3d8","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 13, Iss 9, p 631 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms13090631","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13090631","pdf_url":null,"source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W2057817591","https://openalex.org/W2062176343","https://openalex.org/W2102822087","https://openalex.org/W4292806204"],"related_works":["https://openalex.org/W3022590230","https://openalex.org/W657485438","https://openalex.org/W164200052","https://openalex.org/W3216118368","https://openalex.org/W1483355072","https://openalex.org/W3022244109","https://openalex.org/W4388677443","https://openalex.org/W2337243047","https://openalex.org/W579400037","https://openalex.org/W4254401164"],"abstract_inverted_index":{"We":[0],"have":[1],"derived":[2],"alternative":[3],"closed-form":[4],"formulas":[5],"for":[6],"the":[7,16,20,24,37],"trigonometric":[8,43],"series":[9,44],"over":[10,47],"sine":[11],"or":[12],"cosine":[13],"functions":[14],"when":[15],"immediate":[17],"replacement":[18],"of":[19],"parameter":[21],"appearing":[22],"in":[23],"denominator":[25],"with":[26],"a":[27,33],"positive":[28],"integer":[29],"gives":[30],"rise":[31],"to":[32,45],"singularity.":[34],"By":[35],"applying":[36],"Choi\u2013Srivastava":[38],"theorem,":[39],"we":[40],"reduce":[41],"these":[42],"expressions":[46],"Hurwitz\u2019s":[48],"zeta":[49],"function":[50],"derivative.":[51]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2025-12-22T23:10:17.713674","created_date":"2025-10-10T00:00:00"}