I'm trying to simplify the expression
$$\left|\sqrt{a^2+ibt}\right|^2$$
where $a,b,t \in \Bbb R$.
I know that by definition
$$\left|\sqrt{a^2+ibt}\right|^2 = \sqrt{a^2+ibt}\left(\sqrt{a^2+ibt}\right)^*$$
But how do you find the complex conjugate of the square root of a complex number? And what is the square root of a complex number (with arbitrary parameters) for that matter?
Answer
For any complex number $z$, and any square root $\sqrt{z}$ of $z$ (there are two), we have $$\bigl|\sqrt{z}\bigr|=\sqrt{|z|}$$ Therefore $$\bigl|\sqrt{a^2+ibt}\bigr|^2=\sqrt{|a^2+ibt|^2}=|a^2+ibt| = \sqrt{a^4+b^2t^2}$$
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