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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Complex oscillation and nonoscillation results


Authors: Janne Heittokangas, Katsuya Ishizaki, Ilpo Laine and Kazuya Tohge
Journal: Trans. Amer. Math. Soc. 372 (2019), 6161-6182
MSC (2010): Primary 34M10; Secondary 30D35, 34M03
DOI: https://doi.org/10.1090/tran/7931
Published electronically: August 5, 2019
MathSciNet review: 4024517
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Abstract | References | Similar Articles | Additional Information

Abstract: For an entire coefficient $ A(z)$, classifying the oscillation of solutions $ f$ of the linear differential equation $ f''+A(z)f=0$ has been a long-standing problem since the early 1980s. New results on the following three typical questions are proved: Under which conditions on $ A(z)$ does there exist a solution $ f$ such that

(Q1) $ f$ has no zeros,$ \quad $ (Q2) $ \lambda (f)\geq \sigma (A)$,$ \quad $ (Q3) $ \lambda (f)=\infty $?
Here $ \lambda (g)$ and $ \sigma (g)$ denote the exponent of convergence of the zeros of $ g$ and the order of growth of $ g$, respectively. Several nontrivial examples are given.

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Additional Information

Janne Heittokangas
Affiliation: Department of Mathematics, Taiyuan University of Technology, 79 Yingze West Street, Taiyuan 030024, People’s Republic of China
Address at time of publication: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: janne.heittokangas@uef.fi

Katsuya Ishizaki
Affiliation: Faculty of Liberal Arts, The Open University of Japan, Mihama-ku, Chiba, Japan
Email: ishizaki@ouj.ac.jp

Ilpo Laine
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: ilpo.laine@uef.fi

Kazuya Tohge
Affiliation: College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
Email: tohge@se.kanazawa-u.ac.jp

DOI: https://doi.org/10.1090/tran/7931
Keywords: Complex oscillation, exponential polynomial, exponent of convergence, nonoscillation, zero-free solution.
Received by editor(s): September 3, 2017
Received by editor(s) in revised form: October 2, 2018
Published electronically: August 5, 2019
Additional Notes: This research was financially supported by the Academy of Finland, Grant 268009, the discretionary budget (2016) of the President of the Open University of Japan, JSPS KAKENHI Grants 25400131 and 16K05194, and the Vilho, Yrjö, and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
Article copyright: © Copyright 2019 American Mathematical Society