Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a $\mathbb{C}[[z]]_1$-module.
I want to know what are the crossed homomorphisms $\beta: \mathbb{C}[[z]]_1 \rightarrow \mathbb{C}[[z]]$, that is, the functions that satisfy $\beta(h_2\circ h_1)=\beta(h_1)+\beta(h_2)\circ h_1$
This question is related with group cohomology, since the group $H^1(\mathbb{C}[[z]]_1, \mathbb{C}[[z]])$ is the quotient of the crossed homomorphisms over principal crossed homomorphisms.
That principal crossed homomorphisms are of the form $\beta_g (h) = g\circ h - g$. I found that there is a crossed homomorphism $\beta (h) = \ln(h/z)$ that is not principal, so, my conjecture is that the group $H^1(\mathbb{C}[[z]]_1, \mathbb{C}[[z]])$ is generated by that element (as a $\mathbb{C}$-module). Can anyone help me to prove this (probably using group cohomology techniques)? what moreover results can be extracted using these tools?
Thank you!