I am trying to solve using the saddle point method (large a>0): $$I(\alpha)= \int_{-i\pi/2}^{\pi/2}dz\, (1+z^2)e^{-a\cos(z)}$$ So I find that the point I want to expand about is z=0, because $\partial_z\cos(z)=0\implies z=0,n\pi$ So at $z_0=0$, I get $$I(\alpha)=1\int_{-\epsilon}^\epsilon e^{-a(1-z^2/2+...)}\approx e^{-a}\int_{-\infty}^\infty e^{az^2/2}\, dz\approx i\frac{\sqrt{2\pi}}{\sqrt{a}}e^{-a}$$
My question is if this is a valid approach. Mostly, did I correctly choose to expand about z=0. I get confused on which saddle point to select, because I can deform the integral in many ways.
And then if I want $a<0$, would I approach it the same way?