well I need to know whether there are any zeroes of $\sqrt{z}$? so far I guess $0$ is a singular point of the function as it is not analytic at $z=0$, so It has no zeroes am I right? Is it an isolated singularity? please help.
what are the zeroes of $\sqrt{z}$?
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complex-analysis
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2 Answers
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$\sqrt z=0\implies \sqrt z\sqrt z=z=0$
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As pointed out in the comments, it depends on how you define $\sqrt{z}$. If one wants it to be analytic on its entire domain of definition, then take a branch of the logarithm and use that to define the square root off the branch cut as $\sqrt{z}=e^{\frac{1}{2}\log z}$ (or as -$e^{\frac{1}{2}\log z}$, if you prefer). Now, it isn't difficult to see that as $z\to 0$, then $\sqrt{z}\to 0$ in either case, so we can continuously extend our definition of $\sqrt{z}$ to say that $\sqrt{0}=0$, and in that case, can consider $0$ to be the sole zero of $\sqrt{z}$. We just can't analytically extend it that way, since $0$ is a branch point (of order $2$) of $\sqrt{z}$.