Find the complex roots of equation concerning unknown complex number $z$ \begin{equation} z^\mu=r,\quad \mu>0,r\in\pmb{R} \end{equation}
A solution given by a book is to only consider the principal branch of $z$, in my opinion, I think we should consider all the branches.
The following is my solution \begin{equation} \begin{split} z^\mu&=\text{exp}(\mu(\text{Ln}(z)))\\ &=\text{exp}(\mu(\text{ln}(z)+i\text{Arg}(z)+i2k \pi))\\ &=|z|^\mu\text{exp}(\mu(i\text{Arg)(z)+i2k\pi)},\quad -\pi<\text{Arg}(z)\le\pi\\ \end{split} \end{equation}
If $r<0$, then $r=|r|\text{exp}(i\pi)$. Consequently, \begin{equation} \begin{split} &|z|^\mu=|r|\\ &\text{Arg}(z)=\frac{(1-2k)\pi}{\mu} \end{split} \end{equation} Choose those $k$'s such that $-\pi<\frac{(1-2k)\pi}{\mu}\le\pi$.
But the book only considers the principal branch that corresponds to $k=0$. I don't see why other branches are not considered?
Can someone give me an explanation? Thank you