Repulsive one-dimensional two-component gas

The equation of state is defined by following set of integral equations

$$\rho(k) = \frac{1}{2\pi} + \int\limits_{-Q}^QK(k-\varkappa)\rho(\varkappa)\,\frac{d\varkappa}{2\pi},$$ (6)

where the kernel is

$$K(\xi) = %%\frac{4c}{c^2+4\xi^2}-\frac{\pi}{c\ch \frac{\pi x}{c}} 2\int\limits_0^\infty \frac{\cos\xi x}{1+e^{\gamma x}}\,dx$$ (7)

It is possible to express the kernel in terms of $\beta$-function (see Gradstein-Ryzhik for definition). $K(\xi) = -\frac{1}{\gamma}\left(\beta\left(\frac{i\xi}{\gamma}\right) +\beta\left(-\frac{i\xi}{\gamma}\right)\right).$ The density and energy are given by

$$na_{1D} = \int\limits_{-Q}^Q\rho(k)\,dk,$$ (8)

$$\frac{E}{N} = \left[\frac{1}{na_{1D}}\int\limits_{-Q}^Qk^2\rho(k)\,dk \right]\frac{\hbar^2}{2ma_{1D}^2}$$ (9)