Asymptotic Values of Entire Functions of Infinite Order
We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.
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| Published in: | Computational methods and function theory Vol. 23; no. 2; pp. 381 - 392 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2023
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| Subjects: | |
| ISSN: | 1617-9447, 2195-3724 |
| Online Access: | Get full text |
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| Abstract | We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order. |
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| AbstractList | We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order. |
| Author | Hinkkanen, Aimo Miles, Joseph |
| Author_xml | – sequence: 1 givenname: Aimo surname: Hinkkanen fullname: Hinkkanen, Aimo organization: Department of Mathematics, University of Illinois at Urbana-Champaign – sequence: 2 givenname: Joseph surname: Miles fullname: Miles, Joseph email: joe@math.uiuc.edu organization: Department of Mathematics, University of Illinois at Urbana-Champaign |
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| Cites_doi | 10.1007/BF01457182 10.1007/BF03014756 10.24033/bsmf.922 10.1007/BF01140018 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | Primary 30D20 Entire function Asymptotic value Secondary 31A05 |
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| References | CR4 Gross (CR7) 1918; 79 CR6 Iversen (CR8) 1914 CR5 Sire (CR10) 1913; 41 Ahlfors (CR3) 1932; 6 Ahlfors (CR2) 1930; A1 CR9 Ahlfors (CR1) 1929; 32 Valiron (CR11) 1923; 49 CR12 F Iversen (464_CR8) 1914 J Sire (464_CR10) 1913; 41 LV Ahlfors (464_CR2) 1930; A1 LV Ahlfors (464_CR1) 1929; 32 Wilhelm Gross (464_CR7) 1918; 79 464_CR6 464_CR12 464_CR4 464_CR5 LV Ahlfors (464_CR3) 1932; 6 G Valiron (464_CR11) 1923; 49 464_CR9 |
| References_xml | – ident: CR9 – volume: 6 start-page: 1 issue: 9 year: 1932 end-page: 8 ident: CR3 article-title: Über die asymptotischen Werte der meromorphen Funktionen endlicher Ordnung publication-title: Acta Acad. Aboensis Math. et Phys. – ident: CR6 – volume: A1 start-page: 1 issue: 9 year: 1930 end-page: 40 ident: CR2 article-title: Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen publication-title: Acta Soc. Sci. Fenn. Nova Ser. – year: 1914 ident: CR8 publication-title: Rescherches sur les fonctions inverses des fonctions méromorphes – ident: CR5 – ident: CR4 – volume: 79 start-page: 201 year: 1918 end-page: 208 ident: CR7 article-title: Eine ganze Funktion für die jede komplexe Zahl Konvergenzwert ist publication-title: Math. Ann. doi: 10.1007/BF01457182 – volume: 49 start-page: 415 issue: 3 year: 1923 end-page: 421 ident: CR11 article-title: Sur les valeurs asymptotiques de quelques fonctions méromorphes publication-title: Rend. Circ. Math. Palermo doi: 10.1007/BF03014756 – ident: CR12 – volume: 32 start-page: 1 issue: 6 year: 1929 end-page: 15 ident: CR1 article-title: Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung publication-title: Ann. Acad. Sci. Fenn. Ser. A – volume: 41 start-page: 148 year: 1913 end-page: 160 ident: CR10 article-title: Sur la puissance de l’ensemble des points singuliers transcendants des fonctions inverses des fonctions entières publication-title: Bull. Soc. Math. France doi: 10.24033/bsmf.922 – ident: 464_CR4 – ident: 464_CR5 – ident: 464_CR6 doi: 10.1007/BF01140018 – volume: 49 start-page: 415 issue: 3 year: 1923 ident: 464_CR11 publication-title: Rend. Circ. Math. Palermo doi: 10.1007/BF03014756 – volume-title: Rescherches sur les fonctions inverses des fonctions méromorphes year: 1914 ident: 464_CR8 – volume: 32 start-page: 1 issue: 6 year: 1929 ident: 464_CR1 publication-title: Ann. Acad. Sci. Fenn. Ser. A – ident: 464_CR9 – volume: 79 start-page: 201 year: 1918 ident: 464_CR7 publication-title: Math. Ann. doi: 10.1007/BF01457182 – ident: 464_CR12 – volume: 41 start-page: 148 year: 1913 ident: 464_CR10 publication-title: Bull. Soc. Math. France doi: 10.24033/bsmf.922 – volume: A1 start-page: 1 issue: 9 year: 1930 ident: 464_CR2 publication-title: Acta Soc. Sci. Fenn. Nova Ser. – volume: 6 start-page: 1 issue: 9 year: 1932 ident: 464_CR3 publication-title: Acta Acad. Aboensis Math. et Phys. |
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| Snippet | We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the... |
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| StartPage | 381 |
| SubjectTerms | Analysis Computational Mathematics and Numerical Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics |
| Title | Asymptotic Values of Entire Functions of Infinite Order |
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