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Solve equation perturbatively

  1. Jul 8, 2010 #1
    Hi all! I want to perturbatively solve this equation in [tex]\beta[/tex] at second order in [tex]\alpha[/tex]


    I rewrite this formula in this way


    When I try to solve it perturbatively, I obtain


    The result is


    The square root of which gives


    I know that the correct result is


    Clearly, there is something wrong in the second order of my calculus.

    Can anyone please tell me where I'm mistaking? Thanks a lot.
  2. jcsd
  3. Jul 8, 2010 #2


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    Not quite sure what you are doing here, I'm afraid.
    Here's how I would do it.
    Assume a power series expansion as the following:
    We are to determine the k's in orders of alpha.
    Inserting in your equation, and retaining terms of order 0 and 1, we get:

    To order 2, we have:
    Or, we get the equation for k_2:
    Which yields:
    whereby the result follows.
  4. Jul 8, 2010 #3
    You are right. That's the right way to do it. Thank you very much!
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