= Partial-fraction expansion of the hyperbolic cotangent
{title2=$\coth z=1/z+2z\sum_{n\ge1}(z^2+\pi^2n^2)^{-1}$}
The <hyperbolic-sine infinite product> gives, by taking its logarithmic derivative,
$$
\coth z=\frac1z+2z\sum_{n=1}^{\infty}\frac1{z^2+\pi^2n^2}.
$$
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>.
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