Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to Legendre functions

TL;DR

This work analyzes the ODE $\dfrac{d^{2}w}{dz^{2}}=\{u^{2} f(a,z)+g(z)\}w$ with two turning points that coalesce as $a\to a_{0}$ and develops uniform Liouville–Green expansions in terms of parabolic cylinder functions, including explicit error bounds and constructions valid for complex domains. The authors extend Olver’s turning-point theory to coalescing turning points, derive connections among four recessive solutions, and apply the theory to obtain uniform asymptotics for the associated Legendre functions with large $\nu$ and $\mu$, including explicit representations in terms of $U$ and $V$-type LG functions and detailed coefficients. The paper provides practical, computable coefficients and error controls, along with a Cauchy-integral approach to handle expansions at the turning points, and confirms the results with numerical experiments showing high accuracy. The results generalize and unify prior one-term Legendre approximations, extend them to complex arguments, and offer a robust toolkit for asymptotic analysis of special functions and related eigenvalue problems in physics.

Abstract

Linear second-order ordinary differential equations of the form $d^{2}w/dz^{2}=\{u^{2}f(a,z)$ $+g(z)\}w$ are studied for large values of the real parameter $u$, where $z$ ranges over a bounded or unbounded complex domain $Z$, and $a_{0} \le a \le a_{1} < \infty$. The functions $f(a,z)$ and $g(z)$ are analytic in the interior of $Z$. Moreover, $f(a,z)$ has exactly two real simple zeros in $Z$ for $a>a_{0}$ that depend continuously on $a$ and coalesce into a double zero as $a \to a_{0}$. Uniform asymptotic expansions are obtained for solutions in terms of parabolic cylinder functions and their derivatives, together with slowly varying coefficient functions. The coefficients are readily computable and explicit error bounds are provided. The results are then applied to derive new asymptotic expansions for the associated Legendre functions when both the degree $ν$ and the order $μ$ are large.

Figures (5)

• Figure 1: $\zeta$ plane.
• Figure 1: Graphs of $\mathsf{P}^{-\mu}_{\nu}(x)$ (solid curve), $2\mathsf{Q}^{-\mu}_{\nu}(x)/\pi$ (dashed curve), $M(\nu,\mu,x)$ (dotted curve) and $-M(\nu,\mu,x)$ (dash-dotted curve) for $\nu=50$ and $\mu=(1-a^2)^{1/2}u$ ($u=\nu+\frac{1}{2}$) with $a=0.5$.
• Figure 2: $\hat{\xi}$ plane.
• Figure 2: Graph of $\Omega_{4}(\nu,\mu,x)$ for $\nu=50$ and $\mu=(1-a^2)^{1/2}u$ ($u=\nu+\frac{1}{2}$), with $a=0.5$.
• Figure 3: Graph of $\Omega_{4}(\nu,\mu,x)$ for $\nu=50$ and $\mu=(1-a^2)^{1/2}u$ ($u=\nu+\frac{1}{2}$), with $a=0.1$.

Theorems & Definitions (10)

• Remark 1
• Theorem 2.1
• Proof 1
• Lemma 2.2
• Proof 2
• Definition 4.1
• Theorem 4.2
• Theorem 4.3
• Remark 2
• Proof 3