Laplace’s Method is a technique for approximating integrals of the form
\[\int_a^b \exp (N f(x)) dx\]where \(f(x)\) is a twice-differentiable function and \(N\) is large. An extension of the method to the complex plane is called the saddle point approximation or the method of steepest descent.
Suppose \(x^*\) is a global maximizer of \(f(x)\). If we Taylor Series expand about \(x^*\), we get:
\[f(x) = f(x^*) + (x - x^*) \partial_x f(x) \lvert_{x^*} + \frac{1}{2}(x - x^*)^2 \partial_x^2 f(x)\lvert_{x^*} + HOT\]Because \(x^*\) is a global maximizer of \(f(x)\), we know that
\(\partial_x f(x)\lvert_{x^*} = 0\), by virtue of being a fixed point
\(\partial_x^2 f(x) \lvert_{x^*} < 0\), by virtual of being a maximum
Abusing notation slightly for clarity by writing \(\partial_x^2 f(x)\lvert_{x^*}\) as \(\partial_x^2 f(x^*)\), if we plug in the Taylor Series approximation, we have
\[\begin{align*} \int_a^b \exp (N f(x)) dx &\approx \int_{-\infty}^{\infty} \exp (N f(x)) dx\\ &\approx \int_{-\infty}^{\infty} \exp (N f(x^*) + (x - x^*) \partial_x f(x^*) + \frac{1}{2}(x - x^*)^2 \partial_x^2 f(x^*)) dx\\ &=\exp(N f(x^*)) \int_{-\infty}^{\infty} \exp (-\frac{1}{2}(x - x^*)^2 \big\lvert \partial_x^2 f(x^*) \big\lvert) dx\\ &=\exp(N f(x^*)) \sqrt{ \frac{2 \pi}{N \big\lvert \partial_x^2 f(x^*) \big\lvert} } \end{align*}\]where the last step follows from the Gaussian integral.