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

  1. Nov 26, 2013 #1
    Consider the ODE [itex] [/itex] [itex]y''+P(x)y'+Q(x)y=0[/itex].
    If [itex]\stackrel{limit}{_{x→x_{o}}}P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}Q(x)[/itex] converge, can you call [itex]x_{o}[/itex] a 'regular singular point' besides calling it an 'ordinary point'?

    I am saying this because if [itex]\stackrel{limit}{_{x→x_{o}}}P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}Q(x)[/itex] converge, then [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)[/itex] will also converge. And for a second-order linear ODE for which [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)[/itex] converge, [itex]x_{o}[/itex] is termed as a regular singular point.
     
    Last edited: Nov 26, 2013
  2. jcsd
  3. Nov 26, 2013 #2

    lurflurf

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    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.
     
  4. Nov 27, 2013 #3
    I totally get that! Thanks :)
     
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