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[{"key": "dc.contributor.advisor", "value": "Kilpel\u00e4inen, Tero", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Karttunen, Hanna-Kaisa", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2015-01-27T16:36:04Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2015-01-27T16:36:04Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2014", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1465492", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/45160", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on syvent\u00e4\u00e4 tietoja kompleksianalyysist\u00e4 tutustumalla harmonisiin funktioihin ja konformikuvauksiin. Funktioita, jotka toteuttavat Laplacen yht\u00e4l\u00f6n, kutsutaan harmonisiksi funktioiksi. Harmonisten funktioiden m\u00e4\u00e4ritt\u00e4miseen voidaan k\u00e4ytt\u00e4\u00e4 Cauchy-Riemannin yht\u00e4l\u00f6it\u00e4. Harmoniset funktioit ovat yhteydess\u00e4 analyyttisiin funktioihin, sill\u00e4 harmonisten funktioiden avulla voidaan selitt\u00e4\u00e4 analyyttisten kuvausten teoriaa ja p\u00e4invastoin. T\u00e4m\u00e4n tutkielman kannalta t\u00e4rkeimpi\u00e4 analyyttisi\u00e4 kuvauksia ovat injektiiviset kuvaukset, jotka tunnetaan my\u00f6s konformikuvauksina. Konformikuvaukset ovat alueiden v\u00e4lisi\u00e4 kuvauksia, jotka s\u00e4ilytt\u00e4v\u00e4t kulmien suuruuden ja suunnan ja joiden derivaatta on \u00e4\u00e4rellinen ja nollasta eroava. Harmonisten funktioiden ja konformikuvausten v\u00e4lill\u00e4 on monia t\u00e4rkeit\u00e4 yhteyksi\u00e4. Esimerkiksi, jos harmoniselle funktiolle tehd\u00e4\u00e4n konforminen muuttujanvaihto, niin my\u00f6s tuloksena saatu funktio on harmoninen. \n\nTutkielman motivaationa on oppia ratkaisemaan Dirichlet'n ongelma erityisesti puolitasossa, kiekossa ja monikulmiossa. Dirichlet'n ongelma m\u00e4\u00e4ritell\u00e4\u00e4n usein sellaisessa alueessa, jossa se on vaikea ratkaista. Siten tavoitteena on l\u00f6yt\u00e4\u00e4 analyyttinen kuvaus monimutkaisesta alueesta yksinkertaisempaan alueeseen, jossa ongelma on ratkaistavissa. T\u00e4llainen analyyttinen kuvaus l\u00f6ydet\u00e4\u00e4n tunnettujen konformikuvausten joukosta tai se ratkaistaan esimerkiksi lineaaristen rationaalikuvausten tai Schwarz-Christoffelin kaavan avulla. Puolitasossa ja yksikk\u00f6kiekossa ongelman ratkaisemiseen voidaan soveltaa Poissonin integrointikaavoja. Dirichlet'n ongelman ratkaiseminen noudattaa nelj\u00e4n vaiheen ratkaisumenetelm\u00e4\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Hanna-Kaisa Karttunen (hakarttu) on 2015-01-27 16:36:03.610312. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2015-01-27T16:36:04Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201501271190.pdf: 4338181 bytes, checksum: 5d07b64b4e4f35b8e235f5ef2e02ae55 (MD5)\nlicense.html: 4830 bytes, checksum: fe457e4e80bef8420881314340d1359c (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2015-01-27T16:36:04Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201501271190.pdf: 4338181 bytes, checksum: 5d07b64b4e4f35b8e235f5ef2e02ae55 (MD5)\nlicense.html: 4830 bytes, checksum: fe457e4e80bef8420881314340d1359c (MD5)\n Previous issue date: 2014", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (65 sivua)", "language": null, "element": "format", "qualifier": "extent", "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": "harmoninen funktio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "konformikuvaus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Laplacen yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Cauchy-Riemannin yht\u00e4l\u00f6t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "analyyttinen funktio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Dirichlet'n ongelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lineaarinen rationaalikuvaus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Schwarz-Christoffelin kaava", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Poissonin integrointikaava", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Harmoniset funktiot kompleksialueessa ja konformikuvaukset", "language": null, "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-201501271190", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master's 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": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "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": "dc.date.updated", "value": "2015-01-27T16:36:05Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "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": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktiot", "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": "funktioteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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