Laplace's method - Aetherwisp

Laplace's method is a technique used to approximate integrals of the form

\int_{a}^{b}e^{Mf(x)}\ dx

where $M$ is a large constant and $f(x)$ is a twice-differentiable function with a unique global maximum. $a$ and $b$ may be infinite.

Laplace's method can be generalized to work on path integrals over the complex plane. This generalization is known as the saddle point method.

Theory#

Call $x_{0}$ the global maximum of $f(x)$ . We can Taylor expand $f(x)$ around that point and stop at the second order

f(x)\simeq f(x_{0})+f'(x_{0})(x-x_{0})+ \frac{1}{2}f''(x_{0})(x-x_{0})^{2}

but since $x_{0}$ is a stationary point, $f'(x_{0})=0$ , so we get

f(x)\simeq f(x_{0})+ \frac{1}{2}f''(x_{0})(x-x_{0})^{2}

Since $x_{0}$ is a maximum, in that point we have $f''(x_{0})\leq 0$ . For points in the neighborhood of $x_{0}$ , we can approximately state

f(x)\simeq f(x_{0})- \frac{1}{2}|f''(x_{0})|(x-x_{0})^{2}

If we plug this in the integral we get

\int_{a}^{b}e^{Mf(x)}\ dx\simeq e^{Mf(x_{0})}\int_{a}^{b}e^{- \frac{1}{2}M|f''(x_{0})|(x-x_{0})^{2}}\ dx

If we replace $a$ and $b$ with $-\infty$ and $+\infty$ (if they weren't already) as if $M$ is large, the integral far from $x_{0}$ is almost zero, so there is little difference. Doing so makes it into a Gaussian integral (an integral of a function like $e^{-x^{2}}$ ), which can be solved as

\boxed{\int_{a}^{b}e^{Mf(x)}\ dx\simeq \sqrt{ \frac{2\pi}{M|f''(x_{0})|} }e^{Mf(x_{0})}}

which becomes more and more precise as $M\to \infty$ .