$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$

Chapter 3. Asymptotic Solutions of Linear ODEs

Introduction and classification

1. Introduction
2. Classification: ordinary points
3. Classification: regular singular points
4. Classification: irregular singular points
5. Classification: Infinity

Introduction

In this Chapter we consider linear ODEs with analytic coefficients in the complex domain: \begin{equation} a_n (z)u^{(n)}(z)+ a_{n-1} (z)u^{(n-1)}(z)+\ldots+a_1 (z)u'(z)+a_0(z)u(z)=0 \label{eq-3.1.1} \end{equation} where $a_k(z)$ are analytic functions. After division by $a_n(z)$ we get the similar equation albeit with the leading coefficient equal to $1$: \begin{equation} u^{(n)}(z)+ p_{n-1} (z)u^{(n-1)}(z)+\ldots+p_1 (z)u'(z)+p_0(z)u(z)=0 \label{eq-3.1.2} \end{equation} where $p_k(z)$ are also analytic functions.

Remark 1.

1. Analytic does not exclude the presence of singular points.
2. Read in the Complex Variables textbook classification of isolated singular points of analytic functions (poles, essentially singular points). Also read about branching points.
3. We consider such equations rather than more general equations with smooth coefficients because usually one needs equations with analytic coefficients and the theory here is more developed.

Classification: ordinary points

Definition 1. Point $z_0\ne\infty $ is an ordinary point of equation (\ref{eq-3.1.2}) if $p_k(z)$ are analytic at $z_0$ for all $k=0,1,\ldots,n-1$.

We are going to prove in Section 3.2 that then solution $u(z)$ is analytic at $z_0$ and moreover the radius of convergence of its Taylor expansion is at least the distance to the nearest singularity.

Example 1. Consider toy-model: Constant coefficient equations. Check ODE textbook.

Classification: regular singular points

Definition 2. Point $z_0\ne\infty $ (which is not an ordinary point in the sense of Definition 1) is a regular singular point of equation (\ref{eq-3.1.2}) if $p_k(z)(z-z_0)^{n-k}$ are analytic at $z_0$ for all $k=0,1,\ldots,n-1$.

In other words: $z_0$ could be a pole of degree not exceeding $n-k$.

Example 2. Consider toy-model: Euler equation \begin{equation} z^n u^{(n)}(z)+ z^{n-1} q_{n-1} u^{(n-1)}(z)+\ldots+zq_1 u'(z)+q_0 u(z)=0 \label{eq-3.1.3} \end{equation} with constant $q_{n-1},\ldots, q_0$ has solutions of the form $z^\alpha$ where $\alpha$ is an incidical exponent--a root of the incidical equation \begin{equation} \alpha(\alpha-1)\cdots (\alpha -n+1) + q_{n-1}\alpha(\alpha-1)\cdots (\alpha -n+2)+\ldots +q_1 \alpha +q_0=0; \label{eq-3.1.4} \end{equation} if the multiplicity of this root is $r\ge 2$ then there are also solutions $z^\alpha (\ln z)^j$, $j=1,\ldots,r-1$.

The general solution is a linear combination of solutions described above.

Remark 2.

1. $(z-z_0)^\alpha$ is analytic at $z_0$ if and only if $0\le \alpha\in \bZ$;
2. $(z-z_0)^\alpha$ has a pole of degree $-\alpha$ at $z_0$ if and only $0 > \alpha\in \bZ$;
3. Otherwise (that is, for $\alpha \in \bC\setminus \bZ$) $(z-z_0)^\alpha$ has a branching point at $z_0$; the number of branches is finite if and only if $\alpha$ is a real and rational; the number of branches is $s$ where $s$ is the denominator in the irreducible representation of $\alpha=t/s$ with $t,s\in \bZ$, $s>0$;
4. $(z-z_0)^\alpha(\ln (z-z_0))^j$ has a branching point at $z_0$ as $j\ge 1$ and the number of branches is infinite.

Remark 3. In the general case assuming that $\alpha+1$,\ \alpha+2,\ \ldots$ are not incidical exponents¹ we get solutions of the form \begin{equation} \sum|_{0\le k\le j} A_k(z)(z-z_0)^\alpha (\log (z-z_0)^k,\qquad \text{with }\ A_j(z_0)\ne 0 \label{eq-3.1.5} \end{equation} where $A_k(z)$ are analytic functions at $z_0$. Here $j=0,\ldots,r-1$.

The Taylor series of $A_k$, expanded about $z_0$, has a radius of convergence at least as large as the distance to the next nearest singularity.

Classification: irregular singular points

Definition 3. Point $z_0\ne\infty $ is an irregular singular point of if it is neither an ordinary point nor a regular singular point.

Remark 4. There is no comprehensive theory for irregular singular points. What can be said is the following:

1. At least one solution is not of the form of those given previously for ordinary and regular singular points;
2. While it may happen that a solution is analytic, or has a branch point at an irregular point z0, typically every solution has an essential singularity at z0.

Classification: Infinity

Recall that in the theory of Complex Variables $z=\infty\in \bC^*$ (extended complex plane) is treated as an ordinary point $\zeta=0$ after substitution $z=1/\zeta$.

Definition 4. The point $z_0 = \infty$ is:

1. an ordinary point if $\zeta=0$ is an ordinary point of equation obtained after substitution $z=1/\zeta$;
2. a regular singular point if $\zeta=0$ is a regular singular point of equation obtained after substitution $z=1/\zeta$;
3. an irregular singular point if $\zeta=0$ is an irregular singular point of equation obtained after substitution $z=1/\zeta$.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$