Is it possible to obtain an explicit solution for $K$ for the following equation?

$$(e^K - 1)(e^{\beta K} - 1) = q$$ for $0\leq q \leq 4$ and $0\leq \beta \leq 1$

For $\beta=1$ one gets $K=\log{\left(1+\sqrt{q}\right)}$.


Actually this equation determines the critical point of the Potts model on the square lattice with a ratio $\beta$ of horizontal and vertical coupling constants. I was wondering if it is possible to find a transformation of this equation for specific, possibly rational, values of $\beta$, which allows to solve this equation.

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    $\begingroup$ Very unlikely, although I don't have a proof at this time. Note that the equation is equivalent to the (algebraic, when $\beta$ is rational) equation $(u - 1)(u^{\beta} - 1) = q.$ $\endgroup$ – Dave L. Renfro Jul 2 '14 at 20:37

Is it possible to obtain an explicit solution for K for the following equation?

That would depend on the specific value of q. For an unspecified q, the only $\beta\in[0,1]$ for which an explicit formula $($in terms of radicals$)$ exists are $1,\dfrac12$, and $\dfrac13$. Unless you're OK with expressing K in terms of Bring radicals or hypergeometric functions.

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