I am interested in finding the asymptotic approximation of the following integral:
$$ I(x,y) =\int_{-\infty}^{\infty} \frac{k_{2}A}{k_{2}A+B}\frac{e^{i(k_{1}x + k_{2}y)}}{k_{2}}dk_{1}$$
where
$$k_{2}=\sqrt{k^{2} - k_{1}^{2}}$$
and A and B are constants.
So far I defined my function
$$\phi(k_{1})=e^{i(k_{1}x + k_{2}y)}$$
from which I found the saddle points where $$\frac{d\phi(k_{1})}{dk_{1}}=0$$ which I found to be
$$k_{1,SP}=\pm\frac{kx}{r}$$
where $$r=\sqrt{x^{2}+y^{2}}$$
I am stucked in finding the steepest descent path because I don't know how to deal with the possible poles I have in the first expression of the integral. Can anyone give some hints on how to continue working on this derivation? I am interested in not only obtain the leading order term.
Thanks in advance!