Global study on analytic solutions of functional equations

1.研究了具有双周期系数的Riceati方程。证明了所有的周期解都是双周期的，并得到了它们的周期和表达式。研究了具有单周期亚纯系数的线性微分方程解的零点分布。利用解的Stokes现象，给出了一类包含Hill方程和Mathev方程的方程的零频估计。对于Painleve Transcendents(I)、(II)、(IV)，我们获得了增长的估计。对于Painleve Transcendents(III)、(V)关于普遍覆盖，我们得到了增长的估计。对于(I)，我们也得到了一个较低的估计。研究了小目标的价值分布。对于属于PI-族的高阶Painleve方程，我们得到了亚纯解的极点频率的一个较低估计。

1. We have studied Riceati differential equations with doubly periodic coefficients. It was shown that all the periodic solutions are doubly periodic, Also we btained their periods and expressions.2. We have studied the distribution of zero of solutions satisfying linear differential equations with simply periodic meromorphic coefficients. Using Stokes phenomena of solutions, we gave estimates for zero-frequency of solutions of a class of equations containing Hill equations and Mathev equations.3. For Painleve transcendents (I), (II), (IV), we obtained estimates for the growth. For Painleve transcendents (III), (V) on the universal covering, we obtained estimates for the growth. For (I), we obtained a lower estimate as well. We studied the value distribution of small target as well.4. For higher order Painleve equations belonging to PI-hierarchy, we obtained a lower estimate for the frequency of poles of meromorphic solutions.