Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example:

$$f(z) = \sqrt{1-z^2}$$

This function uses the (complex) logarithm to define it:

$$e^{\large \frac{1}{2}\log(1-z^2)}$$

$$e^{\large \frac{1}{2}[\ln|1-z^2| + i\arg(1-z^2)]}$$

And so we can see that the function is not defined at $\pm1$. They are so-called "branch points", and this function requires two branch cuts.

So, my questions are:

a) is every point along the branch cut also called a "branch point", or is it just the "starting point" in the branch cut that is called a branch point?

b) needing two branch cuts, does this mean we have two functions? Way before seeing a function such as this one, we learn that for multi-valued "functions", once we specify a branch, it then becomes a well-defined, single-valued, genuine function. But we usually only make one branch cut, though. Or is the example I gave above just...one function requiring two branch cuts? Then, making two branch cuts, does this mean we have chosen one branch of $f(z) = \sqrt{1-z^2}$?

...it doesn't mean that we have chosen two branches of the function, right?

Thanks,

$\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}$For posterity: Suppose $F$ is a holomorphic function of two complex variables $(z, w)$ that does not vanish identically on any line $\{z = z_{0}\}$.

The equation $F(z, w) = 0$ implicitly defines $w$ as a function of $z$ at each point where $\dd F/\dd w \neq 0$. If $U$ is a non-empty open subset of $\Cpx$, a branch of function defined by $F$ in $U$ is an open set $\overline{U} \subset U \times \Cpx$ and an analytic (i.e., holomorphic) function $f:U \to \Cpx$ satisfying

$(z, f(z)) \in \overline{U}$ for all $z$ in $U$, and for all $(z, w)$ in $\overline{U}$, $F(z, w) = 0$ if and only if $w = f(z)$.

Informally, a sufficiently small piece of the locus $F(z, w) = 0$ is precisely the graph of $f$ over $U$.

A branch point of the locus $F(z, w) = 0$ is a point where $\dd F/\dd w = 0$. If $(z_{0}, w_{0})$ is a branch point, there is a local power series expansion $$0 = F(z, w) = g_{0}(z) + \sum_{k=2}^{\infty} g_{k}(z) (w - w_{0})^{k}$$ with $g_{k}$ holomorphic. If $n$ is the smallest integer greater than or equal to $2$ with $g_{n}(z_{0}) \neq 0$, a short calculation gives $$(w - w_{0})^{n} = -\frac{g_{0}(z)}{g(z, w)},$$ with $$g(z, w) = \sum_{k=0}^{\infty} g_{n+k}(z)(w - w_{0})^{k}$$ holomorphic, and not vanishing at $(z_{0}, w_{0})$. Qualitatively, $w$ is an $n$-valued function of $z$ in every sufficiently small neighborhood of the branch point. A branch point is an "intrinsic" feature of the locus $F(z, w) = 0$.

By contrast, a "branch cut" is an artificial piece of extra data imposed to extract one or more branches in some neighborhood of a branch point. In typical usage, a branch cut for the locus $F(z, w) = 0$ at a branch point $(z_{0}, w_{0})$ is a piecewise-$C^{1}$ arc $\Gamma$ in $\Cpx$ with one end at $z_{0}$, and having the property that

For some non-empty open neighborhood $U$ of $z_{0}$, there exists a branch of function defined by $F$ in $U \setminus \Gamma$, i.e., in some neighborhood of $z_{0}$ with the cut removed.

Particularly,

1. Only the "starting point" of a branch cut is a branch point.

2. (a) The locus $F(z, w) = z^{2} + w^{2} - 1 = 0$ (image below from https://mathoverflow.net/questions/135819) requires only one branch cut; any arc joining the branch points $(1, 0)$ and $(-1, 0)$ will do. (Even removal of the portion of the real axis with $|z| \geq 1$ may be viewed as a single cut passing through $\infty$ in the Riemann sphere.)

(b) This locus has two branches because $F$ is quadratic in $w$, not because there are two branch cuts. With a single branch cut (such as an arbitrary ray from the origin), the $n$th root function $F(z, w) = z - w^{n} = 0$ defines $n$ distinct branches at the origin.