Saddle Point Approximation for the Integral ∫0∞xe-ax-b/√xdx

[derravaragh]

Homework Statement

Apply saddle point approximation to the following integral:

I = ∫₀^∞xe^-ax-b/√xdx a,b > 0

Recall that to derive Stirling formula from the Euler integral in class we required N >> 1. For the integral defined above, identify in terms of a and b appropriate parameter that justifies the use of the saddle point approximation.

Homework Equations

The Attempt at a Solution

In class, my teacher worked through a simpler problem for the integral xⁿe^-x, and started by finding out the maximum x₀=n.

Trying to follow my teachers example, I changed the integral to e^{ln(x) - ax - b/√x}
and looked at the limits. I set f(x) = ln(x) - ax - b/√x, so that as x→∞, f(x)→-∞ and is almost linear, and as x→0, f(x) depends on ln(x) - b/√x.

I guess my issue is I'm not really sure how to determine the parameters. My intuition tells me that a<x₀<b, but I don't know how to show this, or if that's even what is asked of me. I tried to take the derivative of f(x) to determine x₀ and came up with f'(0) = (1/x)-a-(b/2)*x^-3/2 = 0, which isn't making this easier on me. Am I on the right track or am I completely missing the obvious here?

 

[haruspex]

f'(0) = (1/x)-a-(b/2)*x^-3/2 = 0

You don't mean f'(0), and you have a sign wrong.
Let x' be the solution of f'(x) = 0. Express x as x' plus some new variable, z. Plug that into the integrand and approximate for small z. You can use the f'(x')=0 equation to get some cancellation. (You may find you get too much cancellation and you need to include smaller terms, like z².)