Laplace's method  From Wikipedia, the free encyclopedia Jump to:
navigation, search  In mathematics, Laplace's method, named
after Pierre-Simon Laplace, is a technique used to approximate
integrals of the form   \int_a^b\! e^{M f(x)} \, dx  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).  The idea of Laplace's method[edit]  The function
e(x), in blue, is shown on top for M = 0.5, and at the bottom
for M = 3. Here, ƒ(x) = sin x/x, with a global maximum
at x0 = 0. It is seen that as M grows larger, the approximation
of this function by a Gaussian function (shown in red) is
getting better. This observation underlies Laplace's method.
Assume that the function ƒ(x) has a unique global maximum at x0.
Then, the value ƒ(x0) will be larger than other values ƒ(x). If
we multiply this function by a large number M, the ratio between
(x0) and (x) will stay the same