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

Asymptotic expansions and WKB solution

  1. Jun 20, 2006 #1
    let be e an small parameter e<<<1 then if we want to find a solution to the equation:

    [tex] e\ddot x + f(t)x=0 [/tex]

    then we could write a solution to it in the form:

    [tex] x(t)=exp(i \int dt f(t)^{1/2}/e)[a_{0}(t)+ea_{1}(t)+e^{2}a_{2}(t)+......] [/tex]

    My question is if we could apply Borel resummation (or other technique) to give a "sum" for a divergent series in the form:

    [tex]a_{0}(t)+ea_{1}(t)+e^{2}a_{2}(t)+......\rightarrow \int_{0}^{\infty}dxe^{-x}B(t,x,e)dx [/tex]

    With [tex] B(x,t,e)= \sum_{n=0}^{\infty} \frac{a_{n}(t)e^{n} x^{n}}{n!} [/tex]

    the generating function of the coefficient..so we can extend the domain of convergence for the solution not only to the case e--->0 but to every value of e or at least valid when e-->1.:tongue2: :rolleyes:
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

Similar Discussions: Asymptotic expansions and WKB solution
  1. Asymptotic expansion (Replies: 2)

  2. Asymptotic solution (Replies: 1)