= Hyperbolic cotangent generating function for Bernoulli numbers
{title2=$\coth z=z^{-1}+\sum_{j\geq1}\frac{2^{2j}B_{2j}}{(2j)!}z^{2j-1}$}
The defining generating function of the <Bernoulli numbers>, with $B_1=-1/2$, gives this identity of formal <Laurent series> by $\coth z=1+2/(e^{2z}-1)$. Therefore $a\coth(az)-\coth z$ is regular and odd. Integrating it gives the even formal logarithm of $\sinh(az)/(a\sinh z)$. Its even derivatives of order $k$ at zero are $2^k(a^k-1)B_k/k$, while the odd derivatives vanish. This computes the <cyclotomic higher logarithmic derivatives> of symmetric <cyclotomic units> without introducing a spurious linear term.
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