Can I Choose Any Branch for the Derivative of sqrt(z) in the Principal Branch?

[kcuf]

I understand the basic definitions and ideas behind choosing branches of multivalued functions, but I still have some uncertainty when dealing with multivalued functions. As an example, if I'm in the the principal branch, when can I say that $(z-1)^\frac 1 2(z+1)^\frac 1 2= (z^2-1)^\frac 1 2$; or, what happens when I differentiate $\sqrt z$ in the principal branch, the resulting derivative is $z^{-\frac 1 2}$, am I free to choose whatever branch I want for this derivative, or does it have to correspond to that of the original function.

If anyone could help me clarify this, I would greatly appreciate it.

Thanks

 

[kcuf]

also, maybe I'm just an idiot, but I how do I get this forum to display tex?

 

[kcuf]

Maybe I should clarify what's motivating this question:
I'm trying to use the Christoffel-Schwarz, and the text I have been using says that the powers are computed with respect to the principal branch, however examples that I have seen online just seem to naively combine squareroots together in order to find a potential antiderivative, for example http://planetmath.org/encyclopedia/ExampleOfSchwarzChristoffelTransformation.html . What also troubles me with this example, is if we were restricted to the principal branch, then $\sqrt{z^2-1}$ is not analytic on the upper half plane (because it sends most of the imaginary axis to the negative real axis), so it shouldn't be an antiderivative unless we can choose a different branch for the derivative (which seems wierd), right?