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The free energy of the XY model for $ N \rightarrow \infty $ given by:

$$ F = -\lim_{N\rightarrow\infty}\dfrac{1}{\beta N} \ln Z = -\dfrac{1}{\beta}\ln[2\pi I_0(\beta J)] $$

where $I_n(z) = \int_0^{2\pi} \dfrac{d\phi}{2\pi} e^{z\cos \phi}\cos n\phi$.

How do we compute the high and low temperature limits of F?

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Here you could use the asymptotic forms of the Bessel functions, and consider limits $\beta=\frac{1}{k_BT}\rightarrow 0$ and $\beta=\frac{1}{k_BT}\rightarrow +\infty$.

I suggest that you get hold of copies (or pdfs) of Abramovitz&Stegun and Gradshtein&Ryzhik - indespensable for working with special functions.

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