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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 (x0) and (x) will stay the same (since (x0)/(x) = ƒ(x0)/ƒ(x)), but it will grow exponentially in the function (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 .


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