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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kantinkoski, Sanja", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-05-27T13:04:10Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-05-27T13:04:10Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/75996", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on k\u00e4sitell\u00e4 konformikuvauksia ja konformisia automorfismeja. Tutkielmassa esitell\u00e4\u00e4n erilaisia kuvausongelmia niin kompleksitasossa kuin Riemannin pallollakin. Konformisuus m\u00e4\u00e4ritell\u00e4\u00e4n monin eri tavoin riippuen kirjallisuudesta. T\u00e4ss\u00e4 tutkielmassa k\u00e4yt\u00e4mme m\u00e4\u00e4ritelm\u00e4\u00e4, jonka mukaan analyyttinen injektio on konformikuvaus kunhan m\u00e4\u00e4rittelyjoukko on avoin ja ep\u00e4tyhj\u00e4. Kuvausongelmia ratkaistaan konformikuvauksin ja havainnollistetaan kuvin.\n\nJotta Riemannin pallolla m\u00e4\u00e4riteltyj\u00e4 kuvauksia voitaisiin k\u00e4sitell\u00e4, tulee m\u00e4\u00e4ritell\u00e4 my\u00f6s Riemannin pallo. Riemannin pallo on toisin sanoen laajennettu kompleksitaso, joka koostuu kompleksitasosta ja \u00e4\u00e4ret\u00f6npisteest\u00e4. \u00c4\u00e4ret\u00f6npiste saadaan mukaan liitt\u00e4m\u00e4ll\u00e4 kompleksitasoon pallo, jonka pohjoisnapa k\u00e4ytt\u00e4ytyy kuten \u00e4\u00e4ret\u00f6n suhteessa kompleksitasoon. Kuvausongelmia tarkastellessa huomataan, ett\u00e4 konformiset bijektiot voidaan hajottaa yksinkertaisempien konformisten bijektioiden yhdistelm\u00e4ksi. Esimerkiksi eksponenttifunktion yleinen haara kuvaa konformisti sektorin alueeksi, jota rajoittaa kaksi logaritmista k\u00e4yr\u00e4\u00e4. Paloiteltuna, ensin sektori kuvautuu reaaliakselin suuntaiseksi kaistaleeksi. Sitten kaistale k\u00e4\u00e4ntyy ja lopulta eksponenttifunktio rajoittaa alueen kahdella logaritmisella k\u00e4yr\u00e4ll\u00e4. Eksponentiaalifunktioiden lis\u00e4ksi tarkastellaan rationaalifunktiota, sinifunktiota ja neli\u00f6juurifunktiota sek\u00e4 elliptisi\u00e4 integraaleja.\n\nYksi keskeisimmist\u00e4 tutkielman asioista on M\u00f6bius-kuvaukset. Ne ovat laajennetun kompleksitason eli Riemannin pallon konformikuvauksia. T\u00e4llaiset kuvaukset ovat luonteeltaan geometrisia eli ne kuvaavat ympyr\u00e4t ympyr\u00f6iksi tai suoriksi, jotka tulkitaan \u00e4\u00e4rett\u00f6myyspisteen kautta kulkeviksi ympyr\u00f6iksi. M\u00f6bius-kuvaukset ovat siis kuvauksia Riemannin pallolta itselleen. Jokainen M\u00f6bius-kuvaus on yksinkertaisempien M\u00f6bius-alkeiskuvauksien yhdistetty kuvaus.\n\nLopuksi k\u00e4sitell\u00e4\u00e4n M\u00f6bius-kuvauksiin liittyv\u00e4 sovellus Steinerin porismi. Se on matemaatikko Jakob Steinerin mukaan nimetty kuvausongelma, joka k\u00e4sittelee sis\u00e4kk\u00e4isi\u00e4 toisiaan leikkaamattomia erikeskisi\u00e4 ympyr\u00f6it\u00e4. Steinerin porismille esitet\u00e4\u00e4n ratkaisu, joka k\u00e4ytt\u00e4\u00e4 M\u00f6bius-kuvausten kuvausominaisuuksia.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-05-27T13:04:10Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-05-27T13:04:10Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "36", "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": "konformikuvaus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "steinerin porismi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "automorfismi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Konformikuvauksia", "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-202105273246", "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": "kompleksiluvut", "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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