Register to reply

Branches of squareroot

Share this thread:
kcuf
#1
Feb1-11, 09:00 PM
P: 6
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
Phys.Org News Partner Science news on Phys.org
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
kcuf
#2
Feb1-11, 11:21 PM
P: 6
also, maybe I'm just an idiot, but I how do I get this forum to display tex?
kcuf
#3
Feb2-11, 11:38 AM
P: 6
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/E...formation.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?


Register to reply

Related Discussions
Function (squareroot) Calculus & Beyond Homework 1
Integral of squareroot and exponetial Calculus & Beyond Homework 2
Prove the formula of squareroot 2 Calculus & Beyond Homework 2
Limit of a squareroot combination. Calculus & Beyond Homework 2
Integration with squareroot Calculus 11