Partial-fraction expansion of the hyperbolic cotangent ($\coth z=1/z+2z{\sum }_{n\ge 1}(z^2+{\pi }^2{n}^2{)}^{-1}$)

The hyperbolic-sine infinite product gives, by taking its logarithmic derivative,The identity holds away from the poles, with locally convergent sums. For real $z>0$, it makes $(z\coth z-1)/z^2$ positive and strictly decreasing. It is useful for Hartmann flow flux monotonicity and small-argument expansions of hyperbolic functions.