标题： 微分方程在重合转折点和双极点情况下的渐近展开及其在勒让德函数中的应用 显示英文标题

标题： Asymptotic expansions for solutions of differential equations having a coalescing turning point and double pole, with an application to Legendre functions

摘要： 对于大实参数$u$和$\alpha\in[0,\alpha_{0}]$，分析了二阶线性微分方程$d^{2}w/dz^{2}={u^{2}f(\alpha,z)+g(z)}w$的解的渐近行为，其中$\alpha_{0}>0$是固定的。 自变量$z$在复数域$Z$上取值（可能无界），在该域上$f(\alpha,z)$和$g(z)$除在$z=0$处外都是解析的，微分方程在该点有一个正则奇点。 对于$\alpha>0$，函数$f(\alpha,z)$在$z=0$处有一个二阶极点，在$Z$处有一个单零点，当$\alpha\to 0$时，拐点与极点重合。 对于涉及大$u$的贝塞尔函数近似被构造出来，这些近似是渐近展开式，对$z\in Z$和$\alpha\in[0,\alpha_{0}]$具有统一的有效性。 展开系数由简单的递归生成，获得了明确的误差界，简化了以前的结果。 作为应用，导出了具有大次数$\nu$的连带勒让德函数的统一渐近展开式，在复数$z$的无界区域以及阶数$\mu\in[0,\nu(1-\delta)]$的情况下有效，其中$\delta>0$是任意的。 显示英文摘要

摘要： The asymptotic behavior of solutions to the second-order linear differential equation $d^{2}w/dz^{2}={u^{2}f(\alpha,z)+g(z)}w$ is analyzed for a large real parameter $u$ and $\alpha\in[0,\alpha_{0}]$, where $\alpha_{0}>0$ is fixed. The independent variable $z$ ranges over a complex domain $Z$ (possibly unbounded) on which $f(\alpha,z)$ and $g(z)$ are analytic except at $z=0$, where the differential equation has a regular singular point. For $\alpha>0$, the function $f(\alpha,z)$ has a double pole at $z=0$ and a simple zero in $Z$, and as $\alpha\to 0$ the turning point coalesces with the pole. Bessel-function approximations are constructed for large $u$ involving asymptotic expansions that are uniformly valid for $z\in Z$ and $\alpha\in[0,\alpha_{0}]$. The expansion coefficients are generated by simple recursions, and explicit error bounds are obtained that simplify earlier results. As an application, uniform asymptotic expansions are derived for associated Legendre functions with large degree $\nu$, valid for complex $z$ in an unbounded domain and for order $\mu\in[0,\nu(1-\delta)]$, where $\delta>0$ is arbitrary.