Editing Laplace's method (section)

==Concept==
[[File:Laplaces method.svg|right|150px|thumb| <math>f(x) = \tfrac{\sin(x)}{x}</math> has a global maximum at <math>x = 0</math>. <math>e^{Mf(x)}</math> is shown on top for <math>M = 0.5</math> and at the bottom for <math>M = 3</math> (both in blue). As <math>M</math> grows, the approximation of this function by a [[Gaussian function]] (shown in red) improves. This observation underlies Laplace's method.]]

Let the function <math>f(x)</math> have a unique [[Maxima and minima|global maximum]] at <math>x_0</math>. <math>M>0</math> is a constant here. The following two functions are considered:

:<math>\begin{align}
g(x) &= Mf(x), \\
h(x) &= e^{Mf(x)}.
\end{align}</math>

Then, <math>x_0</math> is the global maximum of <math>g</math> and <math>h</math> as well. Hence:

:<math>\begin{align}
\frac{g(x_0)}{g(x)} &= \frac{M f(x_0)}{M f(x)} = \frac{f(x_0)}{f(x)}, \\[4pt]
\frac{h(x_0)}{h(x)} &= \frac{e^{M f(x_0)}}{e^{M f(x)}} = e^{M(f(x_0) - f(x))}.
\end{align}</math>

As ''M'' increases, the ratio for <math>h</math> will grow exponentially, while the ratio for <math>g</math> does not change. Thus, significant contributions to the integral of this function will come only from points <math>x</math> in a [[Neighbourhood (mathematics)|neighborhood]] of <math>x_0</math>, which can then be estimated.