Find a branch of the multiple-valued function $(4+z^2)^{\frac{1}{2}}$ that is analytic in the given domain $D$ where
$(a) D = \mathbb{C}$ \ $ \{iy:|y| \geq2\}$
$(b) D = \mathbb{C}$ \ $ \{iy:|y| \leq2\}$
Here is my work:
$(4+z^2)^{\frac{1}{2}}$ = $e^{\frac{1}{2}log(4+z^2)}$.
I first tried the principal branch.
The principal branch is $e^{\frac{1}{2}Log(4+z^2)}$, where $Log$ is the natural logarithmic function of real variables.
For $(a)$, the domain of definition is the whole complex plane except on the imaginary axis where $y \geq2$ and $y\leq -2$
For $(b)$, the domain of definition is the whole complex plane except on the imaginary axis where $ -2 \leq y \leq2$ .
The branch cut for the principal branch $e^{\frac{1}{2}Log(4+z^2)}$ is where $4+z^2$ equals to a nonpositive real value. The principal branch is analytic for the domain in $(a)$,
How should I choose a branch that's analytic for the domain of $(b)$?? Thanks for any help.
Best Answer
$z=\pm2i$ are branch points from which branch cuts of order $2$ "emerge". For part (a), the branch cut must be chosen to be the straight line joining $z=-2i$ to $z=2i$. For part (b), the relevant branch cut has two segments: (i) one is the semi-infinite straight line from $z=2i$ to $z=i\infty$; and (ii)the other is the semi-infinite straight line from $z=-2i$ to $z=-i\infty$. The rest of the answer follows, doesn't it?