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Steepest descent, non-analytic roots

  1. Sep 14, 2008 #1


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    For a physics class, I am supposed to evaluate the following integral
    [tex]I_n(a) = \int_{-\infty}^\infty \mathrm dx \, e^{-n x^2/2 + n a x} \cosh^n(x)[/tex]
    as a function of the real non-zero parameter a, in the limit as [itex]n \to \infty[/itex] using the method of steepest descent. The question adds: "Express the ersult in parametric form as a function of the saddle point position."

    The problem I ran into, is that the extremum position [itex]x_0[/itex] satisfies the equation
    [tex]x_0 - a - \tanh(x_0) = 0[/tex]
    which cannot be solved analytically. So I'm getting a bit confused, whether I should just leave the result
    [tex]I_n(a) \simeq e^{- n x_0^2 + n a x_0} \sqrt{\frac{2\pi}{f''(x_0)}} [/tex]
    where f''(x_0) = tanh(x_0) = x_0 - a, or whether I am missing something here.

    I'd appreciate your input.
  2. jcsd
  3. Sep 14, 2008 #2


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    I've tried some more approaches, but in all of them, the problem is in the log(cosh(x)) that's in the exponent. There is no way it can be rewritten without having [itex]\log\cosh(x_0))[/itex], with x0 the position of the minimum of the exponent, in the final expression.

    I will try to see the TA tomorrow, but your suggestions are still welcome.
    (The exercise was given Thursday afternoon, due on Tuesday).
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