Quasiconformal surgery and linear differential equations
Quasiconformal surgery and linear differential equations
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DOI：
10.1007/s11854-019-0007-9
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发表时间：
2015-10
期刊：
影响因子：
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通讯作者：
W. Bergweiler;A. Eremenko
中科院分区：
文献类型：
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作者：
W. Bergweiler;A. Eremenko
我们描述了一种构造超越整函数的新方法，使得微分方程w“+Aw=0有两个线性独立的解，它们的零点相对较少。特别地，我们解决了Bank和Laine的一个问题，证明了存在大于1/2的任意阶的整函数，使得该微分方程有两个线性独立的解，它们的零点具有有限的收敛指数。我们证明了Bank，Laine，Langley，Rossi和Shenin关于这个问题的部分结果实际上是最好可能的。我们还改进了Toda的一个结果，并表明所得到的估计是最好的。我们的方法是基于无限多个系数A的Schwarzian微分方程组(F)=2A的胶合解。
We describe a new method of constructing transcendental entire functionsAsuch that the differential equationw″ +Aw= 0 has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functionsAof any prescribed order greater than 1/2 such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the estimate obtained is best possible. Our method is based on gluing solutions of the Schwarzian differential equationS(F) = 2Afor infinitely many coefficientsA.