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Saddle Point Approximation

  1. Jan 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Apply saddle point approximation to the following integral:

    I = ∫0xe-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.


    2. Relevant equations



    3. 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 eln(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?
     
    Last edited by a moderator: Jan 24, 2014
  2. jcsd
  3. Jan 24, 2014 #2

    haruspex

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    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.)
     
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