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Beyond Saddle point method

  1. Oct 18, 2006 #1
    Beyond "Saddle point" method

    Let's suppose we have an integral:

    [tex] \int_{-\infty}^{\infty}dXe^{-sF(X)}=g(s) [/tex]

    where [tex] X=(x_1,x_2,x_3,....,x_n) [/tex]

    Then for big s we could use "Saddle point method " and all that stuff :rolleyes: :rolleyes: well, we still have the problem in finding a "meaning" for g(s) for every positive s, the idea i had (of course i know it has ben named before :redface: :redface: ) is appart from making the expansion near the "stationary points" so [tex] dF(X)=0 [/tex] is to evaluate in some "wise" points satisfying some other constraint, i.e [tex] dF(X)+F(x)=0 [/tex] or something similar, the idea came to me when examining "Gaussian Cuadrature" where you evalute your function plus a weight function at certain points that are the roots of Laguerre/Hermite?¿ (i can't remember :cry: ) my idea is if something similar can be made to give meaning to the inegral g(s) for s>0 BUT not big (let's say s=1) or for example imposing an extra "multiplier" so the integral becomes:

    [tex] \int_{-\infty}^{\infty}dXe^{-sF(X)-s\lambda g(X)}=g(s) [/tex]
  2. jcsd
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