I have a probably quite simple question RE the HST.
After some work, I obtain as the partition function for the infinite range 1D Ising model $$Z = \int_{-\infty}^\infty \frac{dy}{\sqrt{2\pi / N\beta J}} e^{-N\beta f(y)}$$ where $$f(y)= \frac{J}{2} y^2 - \frac{1}{\beta} \ln[2\cosh(\beta(h + Jy))]$$
Now I want to evaluate the integral in the thermodynamic limit and thus I will be using the method of steepest descent. I am asked to show that $$Z = \sum_i e^{-\beta N f(y_i)}$$ and have to find the equation satisfied by the $y_i$.
So of course my idea was to say that I will get a contribution from each of the local minima of $Z$, which leads to the condition $$\frac{\partial f}{\partial y}|_{y_i} = 0$$ and $$\frac{\partial^2 f}{\partial y^2}_{y_i} > 0$$ However, to obtain the exact form I am asked to show, I'd also need the condition that $$\frac{1}{2} \frac{\partial^2 f}{\partial y^2}|_{y_i} = J$$ because otherwise the prefactors don't come out correct to cancel each other. And now I just don't see how I get this result since the second derivative of $f$ becomes rather messy.