Hyperbolic cotangent generating function for Bernoulli numbers ($\coth z=z^{-1}+{\sum }_{j\ge 1}\frac{2^{2j}{B}_{2j}}{(2j)!}{z}^{2j-1}$)

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.