The saddle point approximation (SPA), a.k.a. stationary phase approximation, is a widely used method in quantum field theory (QFT) and related fields. Suppose we want to evaluate the following integral in the limit $\zeta \rightarrow \infty$ : \begin{equation}\label{eq:integral} {\mathcal I} = \lim_{\zeta \rightarrow \infty} \int_{-\infty}^{\infty} {\mbox d}x \; {\mbox e}^{- \zeta f(x)}. \end{equation}The saddle point approximation can be applied if the function $f(x)$ satisfies certain conditions. Assume that $f(x)$ has a global minimum $f(x_0)=y_{min}$ at $x=x_0$ , which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those. Consider the Taylor expansion of $f(x)$ about the point $x_0$ : \begin{equation}\label{eq:taylor} f(x) = f(x_0) + \partial_x f(x){\Big |}_{x=x_0}(x-x_0) + \frac{1}{2} {\partial_x}^2 f(x){\Big |}_{x=x_0}(x-x_0)^2 + O(x^3). \end{equation}Since $f(x_0)$ is a (global) minimum, it is clear that $f'(x_0)=0$ . Therefore $f(x)$ may be approximated to quadratic order as \begin{equation} f(x) \approx f(x_0) + \frac{1}{2} f''(x_0) (x-x_0)^2. \end{equation}The above assumptions on the minima of $f(x)$ ensure that the dominant contribution to () in the limit $\zeta \rightarrow \infty$ will come from the region of integration around $x_0$ :

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