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[{"key": "dc.contributor.advisor", "value": "Lehtonen, Ari", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Haasianlahti, Ivar", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-26T06:33:19Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-26T06:33:19Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64856", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 tutkielma k\u00e4sittelee reaalisia ja kompleksisia tieintegraaleja tasossa. Kiinnostuksen kohteena ovat erityisesti ne erikoistapaukset, joissa integrointiin liittyv\u00e4 tie on suljettu, ja integroitava kuvaus on joko lokaalisi integroituva vektorikentt\u00e4 tai analyyttinen kompleksimuuttujan kompleksiarvoinen funktio. Tutkielman ytimess\u00e4 ovat kysymykset t\u00e4llaisten tieintegraalien h\u00e4vi\u00e4misest\u00e4. Analyyttisyyden ja lokaalin integroituvuuden m\u00e4\u00e4ritelm\u00e4t luovat pohjan Cauchyn lauseen nelj\u00e4lle versiolle, ja kukin n\u00e4ist\u00e4 lauseista antaa ehdot, joiden vallitessa sek\u00e4 analyyttisen funktion kompleksinen tieintegraali ett\u00e4 lokaalisti integroituvan vektorikent\u00e4n reaalinen tieintegraali (yli suljetun tien) ovat arvoltaan 0. \n\nCauchyn lauseen versiot esitet\u00e4\u00e4n nousevassa j\u00e4rjestyksess\u00e4, eli jokainen versio on seuraajansa erikoistapaus. Ensimm\u00e4inen versio olettaa, ett\u00e4 tien kuvajoukko m\u00e4\u00e4rittelee vy\u00f6hykkeen, eli riitt\u00e4v\u00e4n \"siistin\" tason osajoukon. Toinen versio asettaa topologiset ehdot, joiden vallitessa lokaalisti integroituvalle kent\u00e4lle voidaan konstruoida potentiaalifunktio, joka on l\u00e4heisess\u00e4 yhteydess\u00e4 analyyttisen funktion primitiivin (eli kompleksisen antiderivaatan) k\u00e4sitteeseen. \n\nEdelleen lauseen kolmas versio vaatii, ett\u00e4 integroitava kuvaus on lokaalisti integroituva tai analyyttinen yhdesti yhten\u00e4isess\u00e4 (eli \"rei\u00e4tt\u00f6m\u00e4ss\u00e4\") joukossa. Lopuksi Cauchyn lauseen nelj\u00e4s versio paljastaa tieintegraalien (lineaarikombinaatioiden) h\u00e4vi\u00e4v\u00e4n, jos tie (teiden lineaarikombinaatio) ei \"kierr\u00e4\" yht\u00e4k\u00e4\u00e4n pistett\u00e4, jossa integroitava kuvaus ei ole lokaalisti integroituva tai analyyttinen. Lause yleist\u00e4\u00e4 edelt\u00e4jiens\u00e4 sanoman yhdist\u00e4m\u00e4ll\u00e4 potentiaaliteorian kierrosluvun k\u00e4sitteeseen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-06-26T06:33:19Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-26T06:33:19Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "79", "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": "polku", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "tieintegraali", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lokaali integroituvuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "analyyttisyys", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "homotopia", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "nollahomologisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Cauchyn lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Cauchyn lause ja potentiaalifunktiot", "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-201906263452", "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", 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"value": "m\u00e4\u00e4ritelm\u00e4t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "vy\u00f6hykkeet", "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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