All admissible meromorphic solutions of Hayman's equation
All admissible meromorphic solutions of Hayman's equation
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DOI：
10.1093/imrn/rnu218
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发表时间：
2014-11
影响因子：
1
通讯作者：
R. Halburd;Jun Wang
中科院分区：
文献类型：
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作者：
R. Halburd;Jun Wang
我们找到方程 $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$ 的所有非有理亚纯解，其中 $\alpha$、$\beta$ 和 $\gamma$ 是 $z$ 的有理函数。这样做，我们通过证明所有此类解都具有有限阶来回答 Hayman 问题。除了系数函数的特殊选择外，通解不是亚纯的，并且包含可移动分支对于系数函数的某些选择，方程允许使用非有理亚纯解的单参数族来表明所有此类解都已找到，并允许我们避免因共振可能发生在任意高阶这一事实而产生的问题，我们实际上解决了寻找 Nevanlinna 理论意义上的所有亚纯解的更一般问题，其中系数 $\alpha$、$\beta$ 和$\gamma$ 是亚纯函数。
We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices for the coefficient functions the equation admits a one-parameter family of non-rational meromorphic solutions. Nevanlinna theory is used to show that all such solutions have been found and allows us to avoid issues that can arise from the fact that resonances can occur at arbitrarily high orders. We actually solve the more general problem of finding all meromorphic solutions that are admissible in the sense of Nevanlinna theory, where the coefficients $\alpha$, $\beta$ and $\gamma$ are meromorphic functions.