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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.

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# Steepest descent, non-analytic roots

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