Find Continuity of f_α Along Negative X-Axis

[Samuelb88]

Homework Statement

Find a branch of w=z^{1/2} which is continuous along the negative x-axis.

The Attempt at a Solution

The book proves that the principle square root function |z|^{1/2} \big( \cos(\theta/2) + i \sin(\theta/2) \big), where -\pi < \theta \leq \pi is discontinuous along the negative x-axis.

I've defined a new branch of the square root function f_\alpha (z) = |z|^{1/2} \big( \cos(\theta/2) + i\sin(\theta/2) \big), where \alpha < \theta \leq \alpha + 2\pi. I know the principle square root function is discontinuous along the negative x-axis because the limit as the principle square root function is path dependent as (r,\theta) approaches an arbitrary negative number r_0 e^{i \theta_0}. What's the best way to proceed from here? Should I choose an \alpha such that f_\alpha can only approach the negative x-axis one way?

 

[Dick]

Homework Statement

Find a branch of w=z^{1/2} which is continuous along the negative x-axis.

The Attempt at a Solution

The book proves that the principle square root function |z|^{1/2} \big( \cos(\theta/2) + i \sin(\theta/2) \big), where -\pi < \theta \leq \pi is discontinuous along the negative x-axis.

I've defined a new branch of the square root function f_\alpha (z) = |z|^{1/2} \big( \cos(\theta/2) + i\sin(\theta/2) \big), where \alpha < \theta \leq \alpha + 2\pi. I know the principle square root function is discontinuous along the negative x-axis because the limit as the principle square root function is path dependent as (r,\theta) approaches an arbitrary negative number r_0 e^{i \theta_0}. What's the best way to proceed from here? Should I choose an \alpha such that f_\alpha can only approach the negative x-axis one way?

There's a lot of choices for alpha. Why not pick alpha=0? Where is the discontinuity now?

 

[Samuelb88]

Along the positive x-axis. I understand how I should approach the problem now. Nonetheless, thank you for your help! My book hid the fact that branches are discontinuous along their branch cuts at the end of an example.