11 - Ordinary and Singular Points

Ordinary Point

Consider $n$th order homogenous ODE

$x_0$ is an Ordinary Point if

$$P_j(x)\text{ are analytic in a neighbourhood of }x=0\text{ for all }j$$

In other words each $P_j(x)$ have a convergent power series expansion of form

$$\sum _{k=0}^{\infty }{c}_k(x-x_0{)}^k$$

Properties of Solutions around Ordinary Points

Consider $n$th order homogenous ODE

1. All $n$ linearly independent solutions of the ODE are analytic in a neighbourhood of $x=x_0$
   Hence

$$y(x)=\sum _{k=0}^{\infty }{a}_k(x-x_0{)}^k$$

2. Radius of Convergence of Series Solution $\ge$ distance (in $\mathbb{C}$) to nearest singular point of coefficient functions $P_j(x)$

Note that to find $a_k$ plug expansion into ODE and use power series expansion for each $P_j$ then equate coefficients of powers of $x$ to $0$ then solve for $a_k$ recursively

Singular Points

Consider $n$th order homogenous ODE

$x_0$ is an Singular Point if

$$\text{exists }{P}_j(x)\text{ such that }{P}_j(x)\text{ is not analytic at }{x}_0$$

So general solution $y(x)$ may not be analytic at $x_0$ or the derivatives will blow up at $x_0$

Regular Singular Point

Consider $n$th order homogenous ODE

$x_0$ is a Regular Singular Point of ODE if
Coefficients $P_j(x)$ are not all analytic at $x=x_0$ but modified coefficients

$$p_j(x)\equiv P_j(x)(x-x_0{)}^{n-j}\text{ are all analytic at }x=x_0$$

Any Singular Point at $x=x_0$ which isn’t a Regular Singular Point is Irregular Singular Point

Singularity at Infinity

Consider $n$th order homogenous ODE

Consider substitution

$$t=\frac{1}{x},u(t)=y(x)$$

If there is a singularity at $t=0$ then there is a singularity at $x=\infty$

Standard Form at a Regular Singular Point

Let $x=x_0$ be a regular singular point then ODE is in form

$$Ly=y^{\prime \prime }(x)+\frac{p(x)}{(x-x_0)}{y}^{\prime }(x)+\frac{q(x)}{(x-x_0{)}^2}y(x)=0$$

where $p$ and $q$ are analytic and hence can be expanded as a convergent power series

$$p(x)=\sum _{k=0}^{\infty }{p}_k(x-x_0{)}^k,q(x)=\sum _{k=0}^{\infty }{q}_k(x-x_0{)}^k$$