I have a generating function $G(p,t)=\sum^{\infty}_{n=0}p^nP_n(t)$ and want to calculate the probabilities $P_n(t)$ by means of a contour integral $$P_n(t)=\frac{1}{2\pi i}\oint\frac{dp}{p}G(p,t)p^{-n}$$ which is done by using Cauchy's differentiation formula. The Integration is performed over a closed contour on the complex $p$ plane. For the generating function I have the following expression $$G(p,t)=\exp\left(-\frac{1}{2}\bar{n}(t)\arccos^2p\right)$$ where $n(t)$ is a known function. To calculate the contour integral it is recommended to use the saddle point approximation. The result is $$P_n(t)\approx\sqrt{\frac{x^2_s(1-x^2_s)}{2\pi(n-\bar{n}(t))}}\exp\left(-\frac{1}{2}\bar{n}(t)\arccos^2x_s-n\ln x_s\right)$$ where we assume $n\gg 1$ and $\bar{n}(t)\gg 1$ and $x_s$ is the root of the saddle-point equation $$\frac{x_s}{\sqrt{1-x^2_s}}\arccos x_s=\frac{n}{\bar{n}(t)}$$ My question is now how the contour integral is evaluated. The saddle-point equation leads to a maximum as the second derivative of the negative exponent in the contour integral is positive but it should be negative to apply the saddle point approximation. I mean for the saddle point approximation we are searching of a minimum of $f(x)$ in the integral $\int\limits^{\infty}_{-\infty}\exp(-Nf(x))$ but here I would find a maximum.
Anyway which contour should one choose?
I will appreciate any help. Thanks!