Steepest descent, non-analytic roots

1. Sep 14, 2008

CompuChip

Hi,

For a physics class, I am supposed to evaluate the following integral
$$I_n(a) = \int_{-\infty}^\infty \mathrm dx \, e^{-n x^2/2 + n a x} \cosh^n(x)$$
as a function of the real non-zero parameter a, in the limit as $n \to \infty$ 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 $x_0$ satisfies the equation
$$x_0 - a - \tanh(x_0) = 0$$
which cannot be solved analytically. So I'm getting a bit confused, whether I should just leave the result
$$I_n(a) \simeq e^{- n x_0^2 + n a x_0} \sqrt{\frac{2\pi}{f''(x_0)}}$$
where f''(x_0) = tanh(x_0) = x_0 - a, or whether I am missing something here.

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 $\log\cosh(x_0))$, with x0 the position of the minimum of the exponent, in the final expression.