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Prove that there is no entire function $f(z)= \sqrt z$ inverse to the entire function $g(z)=z^2$ by finding a maximal region $G$ in which $f(z)=\sqrt z$ is analytic. Describe $f(z)$ using polar coordinates.

How can I prove it, especially the first part? Please help me.

Thank you in advance.

  • Well the question says to find a maximal region $G$ on which $f$ is analytic. Have you tried this? – Dave Sep 04 '17 at 19:36
  • No. I dont understand it properly. Would you please explain it a bit clearer? –  Sep 04 '17 at 19:38
  • Is it saying something about radius of convergence? –  Sep 04 '17 at 19:41
  • What have you not understood? – Arnab Chattopadhyay. Sep 04 '17 at 19:57
  • This question deals with some similar results: https://math.stackexchange.com/questions/1747720/is-sqrtz-an-analytic-function – Dave Sep 04 '17 at 20:19
  • For $\Re(z) > 0$ you can define $f(z) = \sqrt{z}$ in the obvious way as $f(r e^{it}) = \sqrt{r} e^{it/2}$. Can you extend this to a larger region ? Where is there a problem ? – reuns Sep 04 '17 at 22:55

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For the first question: suppose that such inverse exist. Then $g$ is biholomorphic, but $g(-1)=g(1)$ being a contradiction with the biyective condition

José Alejandro Aburto Araneda
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