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[{"key": "dc.contributor.advisor", "value": "Koskela, Pekka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Saariaho, Ville-Matias", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-12-16T07:14:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-12-16T07:14:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/84433", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 matematiikan pro gradu -tutkielmassa tarkastellaan kompleksianalyysin keinoin polynomikasvuisia kokonaisia funktioita. Polynomikasvuisuus voidaan muotoilla tarkastelemalla funktion f modulia eli itseisarvoa. Jos funktion f modulia voidaan arvioida ylh\u00e4\u00e4lt\u00e4 |f (z)| \u2264 C|z|^n, miss\u00e4 C < \u221e kaikilla |z| \u2265 1, niin t\u00e4ll\u00f6in polynomi C|z|^n rajoittaa funktion f modulia, joten funktio f on v\u00e4ltt\u00e4m\u00e4tt\u00e4 enint\u00e4\u00e4n\nn-asteinen polynomi.\nFunktion f sanotaan olevan kokonainen, jos se on analyyttinen koko kompleksitasossa. Funktion f analyyttisyys voidaan k\u00e4sitt\u00e4\u00e4 suppenavana potenssisarjana,\njossa ei ole negatiivisia potensseja, avoimessa kiekossa B(z0; r) pisteen z_0 suhteen. Kokonaisen analyyttisen funktion f potenssisarjan suppenemiss\u00e4de on \u00e4\u00e4ret\u00f6n.\nTutkielman viisikohtainen p\u00e4\u00e4lause pohjautuu algebran peruslauseeseen, josta jokainen p\u00e4\u00e4lauseen todistettava kohta on johdettavissa. P\u00e4\u00e4lauseen todistuksissa n\u00e4ytet\u00e4\u00e4n ensin, ett\u00e4 kokonainen analyyttinen funktio f on polynomi, mink\u00e4 j\u00e4lkeen\nmuut todistettavat ominaisuudet johdetaan. Algebran peruslause antaa keinon m\u00e4\u00e4\nritt\u00e4\u00e4 n-asteisen polynomin nollakohtien lukum\u00e4\u00e4r\u00e4n, joka saadaan suoraan polynomin asteluvusta. T\u00e4m\u00e4 yksinkertaiselta kuulostava polynomien ominaisuus tuotti\nentisaikojen matemaatikoille harmaita hiuksia, kunnes Carl Friedrich Gauss todisti\nalgebran peruslauseen v\u00e4it\u00f6skirjassaan vuonna 1799. Nyky\u00e4\u00e4n todistuksia algebran\nperuslauseelle on useita. Er\u00e4s eritt\u00e4in lyhyt todistus pohjautuu Liouvillen lauseeseen, joka on t\u00e4m\u00e4n tutkielman p\u00e4\u00e4lauseen er\u00e4\u00e4n kohdan erikoistapaus.\nP\u00e4\u00e4lauseen todistuksissa usein tarkastellaan muunnosta g(z) = f (1/z), joka antaa keinon tarkastella muunnoksen g napoja. Napa voidaan m\u00e4\u00e4ritell\u00e4 analyyttisen\nfunktion potenssisarjan avulla. Navan m\u00e4\u00e4ritelm\u00e4n mukaan potenssisarjassa on \u00e4\u00e4rellinen m\u00e4\u00e4r\u00e4 negatiivisia potensseja. Jos pystyt\u00e4n n\u00e4ytt\u00e4m\u00e4\u00e4n, ett\u00e4 muunnoksella g on napa, niin t\u00e4ll\u00f6in funktio f on polynomi. Tutkielman p\u00e4\u00e4lause siis antaa erilaisia karakterisaatioita polynomeille.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2022-12-16T07:14:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-12-16T07:14:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "31", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Polynomikasvuiset kokonaiset funktiot", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202212165688", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polynomit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktioteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "analyyttiset funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kokonaiset funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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