The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is $$\epsilon(\beta)=R\cosh\beta-\int\frac{d\beta'}{\pi\cosh(\beta-\beta')}\log(1+\exp(-\epsilon(\beta'))).$$

A reference for this is Al.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models, Nucl Phys B342 (1990) 695-720.

I am interested in the asymptotics of the solutions as $R\rightarrow 0$. Zamolodchikov gives a heuristic argument that for small $R\cosh\beta$ we can neglect the first term. Therefore, the $\beta\rightarrow\beta+const$ invariance is restored and the solution $\epsilon(\beta)$ becomes independent of $\beta$ for small $R\cosh\beta$. Can one deduce a more precise asymptotic behavior? In particular, I would like to write down the small $R$ corrections.

One can easily show that this integral operator acting on $\exp(-\epsilon)$ maps the ball of radius $(e^R-1)^{-1}$ in $C^0(\mathbf{R})$ to itself. Furthermore, it is a contraction for large $R$ (see e.g. http://arxiv.org/abs/0807.4723, appendix C).

Was this equation studied anywhere in the mathematical literature?