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arXiv · 1907.07984

Exponential polynomials in the oscillation theory

Janne Heittokangas·Katsuya Ishizaki·Ilpo Laine·Kazuya Tohge
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Abstract

Supposing that $A(z)$ is an exponential polynomial of the form $$ A(z)=H_0(z)+H_1(z)e^{ζ_1z^n}+\cdots +H_m(z)e^{ζ_mz^n}, $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric location of the leading coefficients $ζ_1,\ldots,ζ_m$ play a key role in the oscillation of solutions of the differential equation $f''+A(z)f=0$. The key tools consist of value distribution properties of exponential polynomials, and elementary properties of the Phragmén-Lindelöf indicator function. In addition to results in the whole complex plane, results on sectorial oscillation are proved.

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Janne Heittokangas, Katsuya Ishizaki, Ilpo Laine, Kazuya Tohge. 2019-07-18. Exponential polynomials in the oscillation theory. https://arxiv.org/abs/1907.07984

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