Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Regular singular points (definition)

  1. Sep 26, 2009 #1

    I am trying to understand the definition of regular point, regular singular point and irregular point

    for example, the ode. what would be the r,rs or i points of this?


    dividing gives the standard form

    y''+(3/x)y' + (4/x^2)y=0

    So, obviously x can't equal zero, does that make x a regular singular point because x=0 gives rise to a singularity? If so, what does "regular" mean?

  2. jcsd
  3. Sep 27, 2009 #2
    it seems you divided wrong, or copied the order of the derivatives wrong, or something.

    Yes, 0 is a singular point since at least one coefficient has a pole at 0.

    In x^3y'''(x)+3x^2y''(x)+4xy(x)=0 it is a regular singular point ... After dividing we have y'''(x)+(3/x)y''(x)+(4/x^2)y(x)=0 and the order of the pole goes up like this: 0,1,0,2 which is lower than the maximum 0,1,2,3 ...

    An example irregular singular point: y'''(x)+(3/x^2)y''(x)+(4/x)y(x)=0 now the pole of order 2 in the y'' term is too large.

    The reason for this classification is that at a regular singular point the solutions can be written as series in a nice way. At irregular singular points this usually cannot be done.
  4. Sep 27, 2009 #3
    Thanks for the reply,

    I copied the ODE right, just didn't divide write, good catch though :)

    I'm not sure what you mean by "poles".

    If something is analytic, it means it can be represented by a series solution, correct?

  5. Sep 28, 2009 #4
    The idea of "poles" comes from complex analysis. In case of a quotient of polynomials (in lowest terms) the poles are the zeros of the denominator.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook