Steepest descent contour includes singularity (asymptotic expansions)

1. Jan 23, 2010

Jerbearrrrrr

1. The problem statement, all variables and given/known data
We require an asymptotic expansion of (t in general complex):
$$\int _{-1} ^\infty \frac{e^{i \lambda t^2} }{\sqrt{1+t}}$$ 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 $$-1+\epsilon$$ 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