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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'd appreciate your input.
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'd appreciate your input.