Consider $f(z) = \sqrt{z^2 + 1}$ with the argument of the square root chosen to be the principal branch $(-\pi, \pi]$. I am tasked with showing that $f$ can be analytically continued along every path in $\mathbb{C}\setminus \{\pm i\}$ and finding all possible germs at $0$. A germ of an analytic function at some generic point $w$ is an equivalence class of elements $(f, D)$ where $(f_1, D_1)\sim (f_2, D_2)$ iff $f_1 \equiv f_2$ in some neighbourhood of $w$, where $D_1, D_2$ are the domains of definition of $f_1, f_2$, respectively.
We know that the branch cuts of the complex square root are obtained in the set $S = \{z \in \mathbb{C}\mid \mathfrak{Re}(z) = 0, \mathfrak{Im}(z)^2 \geq 1\}$. My idea was to take four domains of definitions
$$D_1 = \{z \in \mathbb{C}\mid \mathfrak{Re}(z) > 0\}$$
$$D_2 = \{z \in \mathbb{C}\mid \mathfrak{Im}(z) > 1\}$$
$$D_3 = \{z \in \mathbb{C}\mid \mathfrak{Re}(z) < 0\}$$
$$D_4 = \{z \in \mathbb{C}\mid \mathfrak{Im}(z) < -1\}$$
with corresponding branches of the argument in the square root,
$$\theta_1 \in (-\pi, \pi]$$
$$\theta_2 \in (0, \pi]$$
$$\theta_3 \in (2\pi, 3\pi]$$
$$\theta_4 \in (3\pi, 4\pi]$$
(Question:) Does my approach make sense? I am not confident about my choice of argument branches as I am really, really, bad with branches and branch cuts. Hence the intervals themselves might not make sense althought they might reflect my idea regarding the rotation. If my approach for the analytic continuation does not make sense, how should it be fixed?
Also, supposing that the analytic continuation has been constructed, how do I go about and determine the different germs? My material has zero examples regarding germs of analytic functions, except for the nice and concise definition I had given above.
Thanks!
Best Answer
Think of the composite expression $w=\sqrt{z^2+1}$ as being constructed in two stages: $w=\sqrt{\zeta}$ with $\zeta = z^2+1$. Since the square root always has two values, the expression $w$ is "double-valued". To eliminate the ambiguity (i.e., to create a single-valued continuous solution) one starts initially by constructing a local solution that is single-valued and continuous in a small region around a reference point $z_0$. This local solution is called a germ. Next one tries to extend this local solution to a single-valued continuous function on as large a region in the $z$ plane as possible. Usually this is performed by what is called analytic continuation along a path. (If one gets too greedy, and chooses too large a region, then as one traces the values of $w$ in its plane as $z$ travels on a closed loop path, the values of $w$ need not return to their original value. In such cases we have failed to eliminate the ambiguity, and failed to obtain a single-valued function.
To construct a single-valued function defined on a large region $G$ in the $z$ plane, one usually constructs $G$ by introducing branch cuts.
The standard branch cut for $\sqrt{\zeta}$ is the negative real axis of the $\zeta $plane, say $\zeta = -t$ with $t>0$. To find the corresponding branch cuts in the $z$ plane find all solutions of $z^2+1= -t$. The solutions are $z=\sqrt{-(1+t)} =\pm i \sqrt{1+t}$ which can be sketched as two mirror-symmetric rays in the $z$ plane that both lie on the imaginary axis. One ray travels up from $i$ toward $\infty$ and the other ray travels downward from $-i$. When you delete these branch cuts from the $z$ plane, what remains is a simply-connected region $G$. This ensures that the local solution must extend to a global single-valued solution on $G$. Thus there are two global solutions that we can denote by $w= g_1(z)$ and $w= g_2(z)$ defined on $G$. They satisfy $g_1(z)= - g_2(z)$. Intuitively, we have "disentangled" the two solutions by splitting the solution set into two large pieces called sheets.
It is an interesting advanced topic to explore how the two sheets fuse together along the branch cuts. As you cross a branch cut, one solution smoothly matches up with its opposite solution $g_1(z) \to g_2(z)$. You can make a physical model of this process by making two sheets of clear plastic that are copies of $G$ (slit). Stack them over one another and imagine drawing level contours of $Re(w)$ and $Im(w)$ on these sheets. One set of contours will smoothly join with those of the other sheet as you cross each slit.