Analytical extension of the square root to the whole complex plane In a book, in order to prove a theorem, the authors says ''Let $F \in C^\infty(\mathbb{C})$ such that $F(z) = (1+z)^{1/2}$ for $z \in \mathbb{R}_+$. How can I construct such function? As far as I am concerned, the standard extension arguments only give the existence of an analytic extension for $|z| < 1$.

Note that $C^\infty$ is much weaker than analytic. You can't have an analytic function on $\mathbb{C}$ with the desired properties, but $C^\infty$ is no problem. You can for example take the principal branch of the square root, which is analytic on $\mathbb{C}\setminus (-\infty,-1]$, and a cut-off function that is $\equiv 1$ on a neighbourhood of $[0,+\infty)$ and $\equiv 0$ on a neighbourhood of $(-\infty,-1]$. Multiply the two, and you have (after the continuous extension of the product to $(-\infty,-1]$) a function with the desired property. -- Daniel Fischer

Complex differentiable functions are analytic (once complex differentiable functions, we don't even need to require continuity of the derivative, that follows). $C^\infty$ refers to infinitely often real-differentiable functions, at least conventionally. If one were interested in a complex-differentiable, or even only real-analytic, function with this property, the answer would simply be "doesn't exist". -- Daniel Fischer
