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[{"key": "dc.contributor.advisor", "value": "Kilpel\u00e4inen, Tero", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kauppinen, Jussi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-12-07T13:50:59Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-12-07T13:50:59Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/73002", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Charles Emile Picardin mukaan nimetty Picardin lause ottaa kantaa kompleksisesti differentioituvien eli analyyttisten funktioiden k\u00e4ytt\u00e4ytymiseen. Kyseinen lause on\ntutkielman p\u00e4\u00e4tulos. Tarkalleen lauseessa v\u00e4itet\u00e4\u00e4n, ett\u00e4 kompleksitasossa differentioituva kompleksiarvoinen funktio saa enint\u00e4\u00e4n yht\u00e4 arvoa lukuunottamatta kaikki\narvot. Tutkielmassa tullaan esitt\u00e4m\u00e4\u00e4n lauseelle Harnackin ep\u00e4yht\u00e4l\u00f6\u00f6n perustuva\ntodistus.\nTutkielmassa esitell\u00e4\u00e4n runsaasti tarvittavia esitietoja, jotta lukija voi perehty\u00e4\nhalutessaan huolella p\u00e4\u00e4tuloksen todistuksen taustalla olevaan analyysiin. Esitiedot\nalkavat kompleksisen differentioituvuuden osuudesta, jossa keskityt\u00e4\u00e4n todistamaan\nharmonisten funktioiden kannalta t\u00e4rke\u00e4t Cauchyn ja Riemannin yht\u00e4l\u00f6t. Kompleksisen integroinnin osuudessa rakennetaan todistus Taylorin kehitelm\u00e4lle ja p\u00e4\u00e4tulosta\nmuistuttavalle Liouvillen lauseelle. Molemmat tulokset liittyv\u00e4t olennaisesti harmonisten funktioiden tuloksiin. My\u00f6s M\u00f6bius-kuvausten teoriaa esitell\u00e4\u00e4n sit\u00e4 varten,\nett\u00e4 er\u00e4s Picardin lausetta muistuttava tulos saadaan todistettua harmonisille funktioille. Harmonisten funktioiden teoria liittyy jo suoraan p\u00e4\u00e4tuloksen todistukseen.\nErityisesti t\u00e4rkeit\u00e4 tuloksia kyseisess\u00e4 kappaleessa ovat tulokset, jotka liitt\u00e4v\u00e4t harmoniset funktiot kompleksianalyyttisiin ja reaalianalyyttisiin funktioihin.\nHarnackin ep\u00e4yht\u00e4l\u00f6\u00e4 ja Harnack-funktioita k\u00e4sittelev\u00e4ss\u00e4 kappaleessa esitell\u00e4\u00e4n\nkyseinen ep\u00e4yht\u00e4l\u00f6 ja siihen perustuvat funktiot. Harnack-funktioissa t\u00e4rke\u00e4\u00e4 tutkielman kannalta on niiden yhteys harmonisiin funktioihin. Kappaleessa esitell\u00e4\u00e4n p\u00e4\u00e4tulokseen tarvittava tulos harmonisten funktioiden jonojen k\u00e4ytt\u00e4ytymisest\u00e4. Viimeinen tulos tutkielmassa ennen p\u00e4\u00e4tuloksen todistusta on lemma, jonka avulla Harnackfunktioida voidaan arvioida tarkemmin.\nPicardin lauseelle esitett\u00e4v\u00e4 todistus ei lauseen luonteesta huolimatta juurikaan\nnojaudu kompleksianalyysin tuloksiin. Todistuksessa rakennetaan v\u00e4itteen analyyttisesta funktiosta reaaliarvoisia harmonisia funktioita ja n\u00e4ytet\u00e4\u00e4n, ett\u00e4 ei ole mahdollista, ett\u00e4 alkuper\u00e4inen funktio j\u00e4tt\u00e4\u00e4 saamatta kaksi eri arvoa. Todistuksessa k\u00e4ytet\u00e4\u00e4n siis hy\u00f6dyksi tutkielmassa esiteltyjen funktioiden yhteyksi\u00e4 toisiinsa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2020-12-07T13:50:59Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-12-07T13:50:59Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "25", "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.subject.other", "value": "kompleksianalyysi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "harmoniset funktiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "reaalianalyyttiset funktiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kompleksinen polkuintegraali", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "M\u00f6bius-kuvaukset", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Harnack-funktiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Harnackin ep\u00e4yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Picardin lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Picardin lauseen todistaminen Harnackin ep\u00e4yht\u00e4l\u00f6n avulla", "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-202012076949", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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": 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