On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations

Let xi = root v/u tanh(uv)(-1/2), where u and v are positive integers, and let eta = vertical bar h(xi)vertical bar, where h (t) is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values eta of hyperbolic functions by rationals x/y such that x and y satisfy Diophantine equations. We show that there are infinitely many coprime integers x and y such that vertical bar y eta - x vertical bar << log log y/log y and a Diophantine equation holds simultaneously relating x and y and some integer z. Conversely, all positive integers x and y with y >= c(0) solving the Diophantine equation satisfy vertical bar y eta - x vertical bar >> log log y/log y.