# Stationary phase approximation

1. Oct 25, 2011

### muppet

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

The integral in the first line derives from the radial part of the integral
$$F_n(\mathbf{q})=\int d^2b e^{i\mathbf{q}\cdot\mathbf{b}}(e^{i(b_c/b)^n}-1)$$
where $b_c$ 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.