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,