Let us consider an integral of the form , with parameter *λ*>0, and where *f*(*x*) has a single local minimum in the interval (*a*,*b*). The method of steepest decent, also known as saddle-point integration, is a useful method for approximating such an integral in the small *λ* limit. Namely, if the local minimum occurs at *x*=*x*_{0}, then we can expand *f*(*x*) in the Taylor series about this point; as it is a local minimum, *f*‘(*x*_{0})=0 and *f*”(*x*_{0})≥0. Here, we assume that this second derivative is nonzero. Thus, our Taylor series is

.

Now, we note that as *λ*→0^{+}, the term in the exponent becomes ever more negative, and the integrand approaches zero; the local minimum at *x*_{0} is the “slowest” to approach, and thus the function near that point comes to dominate the rest of the integral; and so

and

Now, we see that the last integral is part of a gaussian integral; the peak of our gaussian is in the region of integration, and as *λ*→0^{+}, the gaussian’s width goes to zero as well, so that the tails become negligible in the limit, and

.

Now, recall that the integral of the gaussian over all real numbers is given by

.

Thus ,

and so

.

Let us perform an example: . This does not fit the form as it is, but can be transformed to do so. First, let us make the substitution . Then we have:

. Second, we note that for positive *u*, .

Thus, we have

,

and the integral above fits our form with and ; thus our approximation will be for *n*→∞. Now, , which is zero for *u*=1. , so , and . Thus, our first saddle-point approximation says

and so

which you may recognize as Stirling’s approximation (see here).

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Tags: Approximation, Factorial, Gamma Function, Integration, Math, Method of Steepest Descent, Monday Math, Saddle-Point Approximation, Saddle-Point Integration, Stirling's Approximation

This entry was posted on January 12, 2009 at 1:29 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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January 15, 2009 at 12:45 pm |

I’m Confused :s