Why Is the Real Part of \( z^{1/2} \) Positive?

[Amer]

as a function $$f(z) = z^{\frac{1}{2}}$$
why the real part is positive
My work
I looked into
$$g(z) = z^2$$ , natural domain is the complex field
we can see that
$$g(z) = g(-z)$$ , g is not 1-1
if $$z = r e^{i\theta}$$
$$-z = e^{i\pi} z = re^{i(\theta + \pi)}$$
so we will restrict the domain to get one-one function so we will have the inverse f
how to restrict it, or how to solve it in another way
Thanks

 

[I like Serena]

as a function $$f(z) = z^{\frac{1}{2}}$$
why the real part is positive
My work
I looked into
$$g(z) = z^2$$ , natural domain is the complex field
we can see that
$$g(z) = g(-z)$$ , g is not 1-1
if $$z = r e^{i\theta}$$
$$-z = e^{i\pi} z = re^{i(\theta + \pi)}$$
so we will restrict the domain to get one-one function so we will have the inverse f
how to restrict it, or how to solve it in another way
Thanks

Hi Amer! :)

The real part does not really have to be positive.
The expression $z^{\frac{1}{2}}$ represents 2 values.
Since:
$$z=re^{i\theta}=re^{i\theta + 2ik\pi}$$
$$z^{\frac 1 2} = (r e^{i\theta + 2ik\pi})^{\frac 1 2}=\sqrt r ~ e^{i \frac \theta 2 + ik\pi}$$
The two distinct values are $\sqrt r ~ e^{i \frac \theta 2}$ and $\sqrt r ~ e^{i (\frac \theta 2 + \pi)}$.

So either $\Re(z^{\frac 1 2}) = \sqrt r ~ \cos(\frac \theta 2)$ or $\Re(z^{\frac 1 2}) = \sqrt r ~ \cos(\frac \theta 2 + \pi)$.

If we pick $k=0$, and also $-\pi < \theta \le \pi$ (the principal argument), and look only at the first value, then the real part will be positive.