Cyclotomic higher logarithmic derivative (${\delta }_k(u)={D^{k-1}(D{f}_u/f_u)|}_{T=0},D=(1+T)d/dT$)

For a norm-compatible sequence of local units, its Coleman power series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ defines these integral additive homomorphisms. In the formal coordinate $z=\log (1+T)$ they are the derivatives of $\log f_u(e^z-1)$; the constant term of the logarithm is irrelevant. A Galois element with cyclotomic value $a$ multiplies the kth derivative by $a^k$. This is the multiplicative-formal-group version of the Coates–Wiles homomorphism.