Isn't Ordinary Point Just a Specific Case of Regular Singular Point?

1. Nov 26, 2013

gikiian

Consider the ODE  $y''+P(x)y'+Q(x)y=0$.
If $\stackrel{limit}{_{x→x_{o}}}P(x)$ and $\stackrel{limit}{_{x→x_{o}}}Q(x)$ converge, can you call $x_{o}$ a 'regular singular point' besides calling it an 'ordinary point'?

I am saying this because if $\stackrel{limit}{_{x→x_{o}}}P(x)$ and $\stackrel{limit}{_{x→x_{o}}}Q(x)$ converge, then $\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)$ and $\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)$ will also converge. And for a second-order linear ODE for which $\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)$ and $\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)$ converge, $x_{o}$ is termed as a regular singular point.

Last edited: Nov 26, 2013
2. Nov 26, 2013

lurflurf

I see what you mean. An ordinary point satisfies the requirements of a regular singular point. For most purposes we want to classify the point (ie ordinary, regular singular, or irregular singular). Thus we would not call a regular singular point an irregular singular point nor an ordinary point a regular singular point.

3. Nov 27, 2013

gikiian

I totally get that! Thanks :)