1. ## Region of analyticity

The region of analyticity of log(sqrt(z)). (using principle branch)

I know the domain of log(z) is the complex plane minus the origin and the negative x-axis (C\{x+iy | x =< 0, y=0}). I'm not sure what domain of sqrt(z) produces this as an image, though. I know z --> sqrt(z) cuts angles in half... sooo would the domain be everything but the positive x-axis? I'm a little confused

2. Because you are using the principal branch of log, you take all the arguments of the numbers sqrt(z) from $(-\pi.\pi]$ , therefore there is no z whose square root sqrt(z) lies on the negative real axis.

So log(sqrt(z)) analicity region is $\mathbb C-\{0\}$

3. Originally Posted by mabruka
Because you are using the principal branch of log, you take all the arguments of the numbers sqrt(z) from $(-\pi.\pi]$ , therefore there is no z whose square root sqrt(z) lies on the negative real axis.

So log(sqrt(z)) analicity region is $\mathbb C-\{0\}$
Ooh okay, I see. I kind of had a feeling it was that but was afraid I was missing something. Thank you!!

4. So you get more idea of whats happening see this applet:

Math 132 Applet 4

choose sqrt as a function and select principal branch.

You will see how there isnt a single point z whose image sqrt lies on the negative real line.

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### region of analyticity

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