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Stationary phase approximation

  1. Oct 25, 2011 #1
    Hi all,
    I'm having difficulty reproducing a stationary phase approximation to an integral.
    I have
    [tex]\begin{eqnarray}
    F_n(y)&=&\int dx x J_0(xy)(e^{ix^{-n}}-1) \\
    &\rightarrow_{large y}&\frac{n^{1/(n+1)}y^{-(n+2)/(n+1)}}{\sqrt{n+1}}\exp[-i(n+1)(y/n)^{\frac{n}{n+1}}]
    \end{eqnarray}[/tex]

    The integral in the first line derives from the radial part of the integral
    [tex]F_n(\mathbf{q})=\int d^2b e^{i\mathbf{q}\cdot\mathbf{b}}(e^{i(b_c/b)^n}-1)[/tex]
    where [itex]b_c[/itex] is a constant and b is the radial coordinate; I made some headway treating the phase here as a vector equation to get an expression for the stationary modulus b in terms of q but I can't reproduce the second line; I'm not entirely sure how to handle the asymptotic behaviour of the Bessel function (which gives me real sin, cos that don't appear in this expression?) but I may be missing something more fundamental, as this approach doesn't give me the factor (n+1) in the exponential.

    Thanks in advance.
     
  2. jcsd
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