# The saddle point approximation

In this appendix we are going to see how the saddle point approximation works in general. Let us define the class of integrals:

$I_{N}=\int e^{Nf(x)}dx\quad \quad N\gg 1$ Suppose that $f(x)$ has a unique maximum at $x=x^{*}$ and that ${\textstyle \lim _{|x|\to \infty }f(x)=-\infty }$ . Then, expanding $f$ around $x^{*}$ we have:
$f(x)=f(x^{*})+{\frac {(x-x^{*})^{2}}{2}}f''(x^{*})+{\frac {(x-x^{*})^{3}}{6}}f'''(x^{*})+\cdots$ (and obviously $f'(x^{*})=0$ because $x^{*}$ is a maximum). Setting $z=x-x^{*}$ we can write, stopping the expansion at the third order:
$I_{N}\approx e^{Nf(x^{*})}\int e^{N{\frac {z^{2}}{2}}f''(x^{*})+N{\frac {z^{3}}{6}}f'''(x^{*})}dz$ Calling $y=z{\sqrt {N}}$ and remembering that ${\textstyle f''(x^{*})<0}$ since $x^{*}$ is a maximum, we have:
$I_{N}=e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|+{\frac {y^{2}}{6{\sqrt {N}}}}f'''(x^{*})}dy$ Therefore for very large $N$ the term proportional to $f'''(x^{*})$ in the exponential (like all the following terms of the complete expansion) is negligible, so:
$I_{N}\approx e^{Nf(x^{*})}{\frac {1}{\sqrt {N}}}\int e^{-{\frac {y^{2}}{2}}|f''(x^{*})|}dy$ and computing the Gaussian integral:
$I_{N}\approx {\sqrt {\frac {2\pi }{N|f''(x^{*})|}}}\cdot e^{Nf(x^{*})}$ Therefore we see that the saddle point approximation essentially states that an integral of the form $I_{N}$ can be approximated, provided that $N$ is large, with the value of the integrand calculated at its maximum (up to a multiplicative factor).