Lemma 1

Let D be the open unit disk in $\mathbf{C}$. There exists a unique holomorphic function $f$ defined on D such that $f(z)^2= 1-z$ for every $z\in D$ and $f(0) = 1$. One has $f^*=f$.

The radius of convergence of the power series

$n=0$

is equal to 1 and its sum $f$ defines a holomorphic function on D (VAR,

R1, p. 27, 3.2.9) taking the value 1 at 0. It satisfies $f(x) =1-x$ for every $x\in D\cap \mathbf{R}$ (FVR, III, p. 19), hence $f(z)^2= 1-z$ for $z\in D$ since the difference $f(z)^2-(1-z)$ is a holomorphic function all of whose successive derivatives vanish at 0 (VAR, R1, p. 27, 3.2.5). By definition, one verifies that $f^*=f$.

Let $g$ be a holomorphic function on D such that $g(z)^2= 1-z$ for every $z\in D$ and such that $g(0) = 1$. The function $g$ does not vanish and the continuous function $f /g$ on D takes its values in $\{-1,1\}$; since D is connected and $f(0) =g$(0), one has $f=g$.