A complex function [tex]f\left(z\right)=\sqrt{z}[/tex] can be splitted into two branches:(adsbygoogle = window.adsbygoogle || []).push({});

1. Principal branch: [tex]f_{1}\left(z\right)=\sqrt{r} e^{i \left(\theta/2\right)}[/tex]

2. Second branch: [tex]f_{2}\left(z\right)=\sqrt{r} e^{i \left[\left(\theta+2\pi\right) /2\right]}[/tex]

My question is, is there a way to show/proof that the principal branch only map the z-plane to the right half plane of w-plane (Re w > 0) to which positive imaginary semiaxis is added, and that the second branch only map the z-plane to the left half plane of w-plane (Re w < 0) to which negative imaginary semiaxis is added??

Then, how exactly is to plot these branches in z-plane and w-plane? I am confused, because r and theta could be anywhere in the z-plane, and hence it could be mapped to any point in the w-plane. Please tell me if what I said is wrong.

Thanks in advance.

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# Mapping of multivalued complex function.

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