Laplace's method

In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form

where ƒ(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b could possibly be infinite. This technique was originally presented in Laplace (1774, pp. 366–367).

Assume that the function ƒ(x) has a unique global maximum at x₀. Then, the value ƒ(x₀) will be larger than the other values of ƒ(x). If we multiply this function by a large number M, the ratio between Mƒ(x₀) and Mƒ(x) will stay the same (since Mƒ(x₀)/Mƒ(x) = ƒ(x₀)/ƒ(x)), but it will grow exponentially in the function ${\displaystyle e^{Mf(x)}}$(see figure).

Thus, significant contributions to the integral of this function will come only from points x in a neighbourhood of x₀, which can then be estimated.

To state and motivate the method, we need several assumptions. We will assume that x₀ is not an endpoint of the interval of integration, that the values ƒ(x) cannot be very close to ƒ(x₀) unless x is close to x₀, and that the second derivative ${\displaystyle f''(x_{0})<0}$.

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