why is each branch of $\log z$ yields a branch of $z^{\alpha}$? $F(z)$ is said to be a $\bf{branch}$ of a multiple-valued function $f(z)$ in a domain D if $F(z)$ is single-valued and continuous in $D$ and has the property that, for each $z$ in $D$, the value $F(z)$ is one of the values of $f(z)$. If $\alpha$ is a complex constant and $z \neq 0$, then we define $z^{\alpha}$ by: $z^{\alpha}= e^{\alpha \log z}$ My textbook says "it is clear from the two definitions above that each branch of $\log z$ yields a branch of $z^{\alpha}$" and gives no more explanation. I'm still confused, can anyone give me a more detailed explanation about why this is the case? Thank you very much.

If $L(z)$ is a branch of $\log z$, then $e^{\alpha L(z)}$ is a branch of $e^{\alpha \log z}$. That's what the textbook means by "yields".