Complex oscillation and nonoscillation results
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- by Janne Heittokangas, Katsuya Ishizaki, Ilpo Laine and Kazuya Tohge PDF
- Trans. Amer. Math. Soc. 372 (2019), 6161-6182 Request permission
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, (Q2) $\lambda (f)\geq \sigma (A)$, (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.References
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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
- MR Author ID: 664068
- Email: janne.heittokangas@uef.fi
- Katsuya Ishizaki
- Affiliation: Faculty of Liberal Arts, The Open University of Japan, Mihama-ku, Chiba, Japan
- MR Author ID: 261275
- 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
- MR Author ID: 109330
- Email: ilpo.laine@uef.fi
- Kazuya Tohge
- Affiliation: College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
- MR Author ID: 253733
- Email: tohge@se.kanazawa-u.ac.jp
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 6161-6182
- MSC (2010): Primary 34M10; Secondary 30D35, 34M03
- DOI: https://doi.org/10.1090/tran/7931
- MathSciNet review: 4024517