0

The question is as follow:

Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, does there exist an anlytic function $g$ defined on H such that $g(z)=\sqrt{f(z)}$, for all $z\in H\cap\mathbb{R}^n$?

For the one variable case, the answer is yes. But can we extend this fact to the general case?

Fuhsuan Ho
  • 540
  • How exactly is this true in the one variable case ? – Hmm. Jun 14 '16 at 11:12
  • @SoumyaSinha Since $f$ is a nowhere vanishing analytic function, $\log f$ exists so you may define $g(z)$ by $$e^{\frac{1}{2}logf(z)}$$ for all $z\in H$. – Fuhsuan Ho Jun 14 '16 at 14:31
  • Suppose $f(z)=z^2+1$. It is positive on the real axis, however it has a zero in $\mathbb{C}$. Thus, your statement that $f$ is nowhere vanishing doesn't seem to be true. – Hmm. Jun 14 '16 at 14:47
  • Oh, thank you! It seems that I need to add "f is nowhere vanishing" in my question to make sure logarithm is well-defined. – Fuhsuan Ho Jun 14 '16 at 14:56
  • See here- http://math.stackexchange.com/questions/74920/the-existence-of-analytical-branch-of-the-logarithm-of-a-holomorphic-function – Hmm. Jun 14 '16 at 15:01
  • Oh, it's a shame that I didn't find this post. Thank you @SoumyaSinha! – Fuhsuan Ho Jun 14 '16 at 15:57

0 Answers0