Let $f:U\to \mathbb{C}\setminus(-\infty,0]$ be a holomorphic function. Does it always have a sqare root function $g^2=f$ which is also analytic in the same region $U$?Note that $U$ represents the open unit disk in $\mathbb{C}$.
I think yes, but am not even able to provide an example of such an $f$. Should we make use of the riemann mapping theorem to produce such a function and its square root. Any hints? Thanks beforehand.
Best Answer
The principal branch of the logarithm is holomorphic in $\mathbb{C}\setminus(-\infty,0]$. Therefore, for any holomorphic function $f:U\to \mathbb{C}\setminus(-\infty,0]$, you can define $$ g: U \to \Bbb C, g(z) = \exp(\frac 12 \log(f(z))) $$ which satisfies $$ g(z)^2 = \exp( \log(f(z))) = f(z) $$ in $U$, i.e. it is a holomorphic square root of $f$.
Note that even more is true: In your case $U$ is the unit disk and that is a simply connected domain. It follows that any non-zero holomorphic function $f$ in $U$ has a holomorphic logarithm and therefore a holomorphic square root.