"If z∈C then what is z2?" was answered by our experts

Branden Valentine

Answered question

2022-02-01

z \in C

then what is

\sqrt{z^{2}} ?

Answer & Explanation

liep3p

Beginner2022-02-02Added 4 answers

Step 1
I'm not sure why specifying $z \in C$ is significant.
If $z = a + b i$
then
$z^{2} = a^{2} + 2 a b i - b^{2}$
Any value (in $C,) \hat{z}$ for which
${\hat{z}}^{2} = a^{2} + 2 b i - b^{2}$
should be a square root of z
The two possible $\hat{z}$ values are
$\hat{z} = a + b i = z$
and
$\hat{z} = - a - b i = - z$

Jordyn Horne

Beginner2022-02-03Added 16 answers

Step 1
I would take the principal square root of a complex number z, to be:
$\sqrt{z} = \sqrt{| z |} (c i s (\frac{1}{2} a r g (z}))$
where $a r g (z)$ is the principal argument of z, which some take to be in $[0, 2 π)$ and others take to be in $(π, π]$
So for radicand $z^{2}$ , I would take
$\sqrt{z^{2}} = \sqrt{| z^{2} |} (c i s (\frac{1}{2} a r g (z^{2})))$
where $\arg (z^{2})$ is the principal argument of $z^{2}$ , which some take to be in $[0, 2 π)$ and others take to be in $(- π, π]$
Given a choice, would prefer $a r g$ in $(- π, π]$

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