Second order linear differential equations with a basis of solutions having only real zeros
Let A be a transcendental entire function of finite order. We show that, if the differential equation w ″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the s...
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| Veröffentlicht in: | Journal d'analyse mathématique (Jerusalem) 2024, Vol.152 (1), p.53-108 |
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| container_title | Journal d'analyse mathématique (Jerusalem) |
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| creator | Bergweiler, Walter Eremenko, Alexandre Rempe, Lasse |
| description | Let
A
be a transcendental entire function of finite order. We show that, if the differential equation
w
″ +
Aw
= 0 has two linearly independent solutions with only real zeros, then the order of
A
must be an odd integer or one half of an odd integer. Moreover,
A
has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values. |
| doi_str_mv | 10.1007/s11854-023-0294-z |
| format | Article |
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A
be a transcendental entire function of finite order. We show that, if the differential equation
w
″ +
Aw
= 0 has two linearly independent solutions with only real zeros, then the order of
A
must be an odd integer or one half of an odd integer. Moreover,
A
has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.</description><identifier>ISSN: 0021-7670</identifier><identifier>EISSN: 1565-8538</identifier><identifier>DOI: 10.1007/s11854-023-0294-z</identifier><language>eng</language><publisher>Jerusalem: The Hebrew University Magnes Press</publisher><subject>Abstract Harmonic Analysis ; Analysis ; Differential equations ; Dynamical Systems and Ergodic Theory ; Entire functions ; Functional Analysis ; Integers ; Mathematics ; Mathematics and Statistics ; Partial Differential Equations ; Topology</subject><ispartof>Journal d'analyse mathématique (Jerusalem), 2024, Vol.152 (1), p.53-108</ispartof><rights>The Author(s) 2023</rights><rights>The Author(s) 2023. This work is published under https://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c311t-b1d4d943434630c932cbe0e39847136f61d0ca9ea772bc0200d561f4ea8314de3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11854-023-0294-z$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11854-023-0294-z$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27922,27923,41486,42555,51317</link.rule.ids></links><search><creatorcontrib>Bergweiler, Walter</creatorcontrib><creatorcontrib>Eremenko, Alexandre</creatorcontrib><creatorcontrib>Rempe, Lasse</creatorcontrib><title>Second order linear differential equations with a basis of solutions having only real zeros</title><title>Journal d'analyse mathématique (Jerusalem)</title><addtitle>JAMA</addtitle><description>Let
A
be a transcendental entire function of finite order. We show that, if the differential equation
w
″ +
Aw
= 0 has two linearly independent solutions with only real zeros, then the order of
A
must be an odd integer or one half of an odd integer. Moreover,
A
has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.</description><subject>Abstract Harmonic Analysis</subject><subject>Analysis</subject><subject>Differential equations</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Entire functions</subject><subject>Functional Analysis</subject><subject>Integers</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Partial Differential Equations</subject><subject>Topology</subject><issn>0021-7670</issn><issn>1565-8538</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp1kE9LxDAQxYMouK5-AG8Bz9VJk6btURb_wYIH9eQhpM10N0tNdpNWcT-9WSp4kmGYw7zfzOMRcsngmgGUN5GxqhAZ5Dx1LbL9EZmxQhZZVfDqmMwAcpaVsoRTchbjBqAoap7PyPsLtt4Z6oPBQHvrUAdqbNdhQDdY3VPcjXqw3kX6ZYc11bTR0UbqOxp9P06btf60bkW9679pwATtMfh4Tk463Ue8-J1z8nZ_97p4zJbPD0-L22XWcsaGrGFGmFrwVJJDm2y1DQLyuhIl47KTzECra9RlmTct5ACmkKwTqCvOhEE-J1fT3W3wuxHjoDZ-DC69VBykFAIqwZKKTao2WYsBO7UN9kOHb8VAHTJUU4YqZagOGap9YvKJiUnrVhj-Lv8P_QA_wXUZ</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Bergweiler, Walter</creator><creator>Eremenko, Alexandre</creator><creator>Rempe, Lasse</creator><general>The Hebrew University Magnes Press</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>2024</creationdate><title>Second order linear differential equations with a basis of solutions having only real zeros</title><author>Bergweiler, Walter ; Eremenko, Alexandre ; Rempe, Lasse</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c311t-b1d4d943434630c932cbe0e39847136f61d0ca9ea772bc0200d561f4ea8314de3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Analysis</topic><topic>Differential equations</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Entire functions</topic><topic>Functional Analysis</topic><topic>Integers</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Partial Differential Equations</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bergweiler, Walter</creatorcontrib><creatorcontrib>Eremenko, Alexandre</creatorcontrib><creatorcontrib>Rempe, Lasse</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal d'analyse mathématique (Jerusalem)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bergweiler, Walter</au><au>Eremenko, Alexandre</au><au>Rempe, Lasse</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Second order linear differential equations with a basis of solutions having only real zeros</atitle><jtitle>Journal d'analyse mathématique (Jerusalem)</jtitle><stitle>JAMA</stitle><date>2024</date><risdate>2024</risdate><volume>152</volume><issue>1</issue><spage>53</spage><epage>108</epage><pages>53-108</pages><issn>0021-7670</issn><eissn>1565-8538</eissn><abstract>Let
A
be a transcendental entire function of finite order. We show that, if the differential equation
w
″ +
Aw
= 0 has two linearly independent solutions with only real zeros, then the order of
A
must be an odd integer or one half of an odd integer. Moreover,
A
has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.</abstract><cop>Jerusalem</cop><pub>The Hebrew University Magnes Press</pub><doi>10.1007/s11854-023-0294-z</doi><tpages>56</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Journal d'analyse mathématique (Jerusalem), 2024, Vol.152 (1), p.53-108 |
| issn | 0021-7670 1565-8538 |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Abstract Harmonic Analysis Analysis Differential equations Dynamical Systems and Ergodic Theory Entire functions Functional Analysis Integers Mathematics Mathematics and Statistics Partial Differential Equations Topology |
| title | Second order linear differential equations with a basis of solutions having only real zeros |
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