# Thread: Complex Analysis: Is Sqrt(z) entire?

1. ## Complex Analysis: Is Sqrt(z) entire?

This question is part of a bigger one, but I need to determine for what domain $\displaystyle \sqrt{z}$ is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.

2. ## Re: Complex Analysis: Is Sqrt(z) entire?

Originally Posted by tarheelborn
This question is part of a bigger one, but I need to determine for what domain $\displaystyle \sqrt{z}$ is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
I feel as though, at least from my experience, this is non-standard terminology. I assume what you means is that you want to find some domain $\displaystyle D\subseteq\mathbb{C}$ for which $\displaystyle \sqrt{z}$ is holomorphic on the entire domain? If so, what if you just take the principal branch $\displaystyle D=\mathbb{C}-(\mathbb{R}-\mathbb{R}^+)$?

3. ## Re: Complex Analysis: Is Sqrt(z) entire?

So I'm basically taking the domain to leave off the negative reals?

4. ## Re: Complex Analysis: Is Sqrt(z) entire?

Originally Posted by tarheelborn
So I'm basically taking the domain to leave off the negative reals?
And zero.

5. ## Re: Complex Analysis: Is Sqrt(z) entire?

Originally Posted by tarheelborn
This question is part of a bigger one, but I need to determine for what domain $\displaystyle \sqrt{z}$ is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
I have a bit of a different take on this question from that of reply #2.
If $\displaystyle f(z)=z^{\frac{1}{2}}=\sqrt r \cos \left( {\frac{\theta }{2}} \right) + \mathbf{i}\sqrt r \sin \left( {\frac{\theta }{2}} \right)$
where $\displaystyle r>0~\&~-\pi<\theta<\pi$ then using the polar form of the C-R equations show the derivative exist.
That is a standard example is many basic textbooks.

6. ## Re: Complex Analysis: Is Sqrt(z) entire?

Thanks; this is a little more along the lines of the way I was taught to address these problems.

7. ## Re: Complex Analysis: Is Sqrt(z) entire?

Originally Posted by tarheelborn
This question is part of a bigger one, but I need to determine for what domain $\displaystyle \sqrt{z}$ is entire. I am not sure how to get started. I suspect I am having a really stupid moment and just can't figure out how to make it fit into Cauchy-Riemann, but I can't work it out. Thanks for your help.
As in...

Entire Function -- from Wolfram MathWorld

... according to the 'standard definition', an f(z) is called entire if it is analytic on the whole complex plane $\displaystyle \mathbb{C}$. That means that in any case the expression 'for what domain $\displaystyle \sqrt{z}$ is entire' is 'selfcontraductory' and the expression 'for what domain $\displaystyle \sqrt{z}$ is analytic' should be used instead of it...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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# IS root z is analytic function

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