Zero distribution of finite order Bank--Laine functions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929651849363456 |
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| author | Zhang, Yueyang |
| author_facet | Zhang, Yueyang |
| contents | It is known that a Bank-Laine function $E$ is a product of two normalized solutions of the second order differential equation $f"+Af=0$ $(\dagger)$, where $A=A(z)$ is an entire function. By using Bergweiler and Eremenko's method of constructing transcendental entire function $A(z)$ by gluing certain meromorphic functions with infinitely many times, we show that, for each $λ\in[1,\infty)$ and each $δ\in[0,1]$, there exists a Bank--Laine function $E$ such that $E=f_1f_2$ with $f_1$ and $f_2$ being two entire functions such that $λ(f_1)=δλ$ and $λ(f_2)=λ$, respectively. We actually provide a simpler construction of the special Bank--Laine functions given by Bergweiler and Eremenko. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_11478 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zero distribution of finite order Bank--Laine functions Zhang, Yueyang Complex Variables Primary 34A20, Secondary 30D15 It is known that a Bank-Laine function $E$ is a product of two normalized solutions of the second order differential equation $f"+Af=0$ $(\dagger)$, where $A=A(z)$ is an entire function. By using Bergweiler and Eremenko's method of constructing transcendental entire function $A(z)$ by gluing certain meromorphic functions with infinitely many times, we show that, for each $λ\in[1,\infty)$ and each $δ\in[0,1]$, there exists a Bank--Laine function $E$ such that $E=f_1f_2$ with $f_1$ and $f_2$ being two entire functions such that $λ(f_1)=δλ$ and $λ(f_2)=λ$, respectively. We actually provide a simpler construction of the special Bank--Laine functions given by Bergweiler and Eremenko. |
| title | Zero distribution of finite order Bank--Laine functions |
| topic | Complex Variables Primary 34A20, Secondary 30D15 |
| url | https://arxiv.org/abs/2312.11478 |