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Steepest descent contour includes singularity (asymptotic expansions)

  1. Jan 23, 2010 #1
    1. The problem statement, all variables and given/known data
    We require an asymptotic expansion of (t in general complex):
    [tex]\int _{-1} ^\infty \frac{e^{i \lambda t^2} }{\sqrt{1+t}} [/tex] dt
    in the limit (lambda) tends to infinity.
    Hint given is to sketch the path of Im(it^2)=const through t=0 and t=-1 in the complex t-plane.

    3. The attempt at a solution

    I have a candidate steepest descent path (it's kind of a standard one - a bit of a hyperbola and a bit of y=-x) but the integral 'starts' at a singularity. What do we do about that?
    Could perhaps try starting the integral from [tex]-1+\epsilon[/tex] but how do we go about doing that in practice? (Since we have to choose a convenient direction for epsilon to tend to -1 from)

    thanks
     
    Last edited: Jan 23, 2010
  2. jcsd
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