I have an integral to evaluate
$I = \int_m^{\infty} \sqrt{x^2-m^2} \;e^{-ixb} dx$
where m and b is constant with b is very large as per the requirement for saddle point approximation. In saddle point method if we given the integral of the form
$I = \int_a^b g(x) \;e^{f(x)\,t} dx$
where t is a constant and very large, the soluation is
$I \approx g(x_0) \,e^{f(x_0)\,t}\int_a^b \;e^{\frac{f''(x_0)\,t\,(x-x_0)^2}{2} } dx$
we expanded the function $f(x)$ about its maxima. $x_0$ is the point where $f(x)$ is maximum.
In this problem $f(x) = -ix$ and $ g(x)= \sqrt{x^2-m^2} $. How do I use saddle point method here? Because when $x=m \;,f(x)$ is maximum and then $g(x_0)=0$. If I am wrong please correct me.