# How performs the function $z^{1/2}$ in the complex plane?

How performs the function $z^{1/2}$ in the complex plane? Thanks for your help I know it's a multi-valued function and that we must be careful with the branch on which it is defined, one of my questions is what is the main branch?, Where is continuous and which is defined?

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Who's they?.... –  lhf May 21 '12 at 0:59
See Riemann surface. –  lhf May 21 '12 at 1:00
I just want to see their behavior in two dimensions –  Daniela del Carmen May 21 '12 at 1:02

It is defined on $\mathbb{C}\backslash{(-\infty, 0]}$ (which is the main branch of the complex Logarithm) and is continuous everywhere.