I have a function $$f(z)=(z-1)^{3/5}(z+1)^{2/5}$$
and I have the branch of this function chosen such that $$-\pi<\arg(z\pm1)\leq\pi$$
How do I show that a branch cut is not required on the section $(-\infty,1)$ of the real axis?
I have defined $(z-1)=r_1e^{i\theta_1}$ and $(z+1)=r_2e^{i\theta_2}$ and thus can get $f(z)$ into the form $$r_1^{3/5}r_2^{2/5}e^{i/5(3\theta_1+2\theta_2)}$$
Now here is where I get confused but I think I need to find $f(x\pm0i)$on the section $(\infty,1)$ of the real axis and determine continuity but I don't know how to do this for this specific example and would appreciate some help. Thank you.
Best Answer
You only need to choose a branch cut so that all of the branch points lie on the cut. Remember that a branch point is a point where the function is discontinuous when traversing an arbitrarily small circuit about the point. By choosing a branch cut like this you're essentially acknowledging this behavior and choosing specifically where this discontinuity will occur.
The points $z = \pm 1$ are clearly branch points, and they are the only finite branch points. The point at infinity may also be a branch point. In this case it isn't (shown below), but if it were then you would need to choose your cut so that it contains all three branch points ($1$, $-1$, and $\infty$). You could do this by choosing it to be the real interval $(-\infty,1]$, but there are infinitely many other choices you could make too.
To see that the point at infinity is not a branch point, rewrite the function as
$$ f(z) = z\,\left(\frac{1}{z}-1\right)^{3/5}\left(\frac{1}{z}+1\right)^{2/5}. $$
Let $|z| > 1$. By doing this, we can ensure that neither of the two quantities $a = \frac{1}{z}-1$ or $b = \frac{1}{z}+1$ will make a circuit about the origin, regardless of what $z$ is doing. They are both restricted to unit disks centered at $-1$ and $1$, respectively. Indeed, $|a+1|<1$ and $|b-1|<1$.
Now, with this condition on $|z|$, if $z$ makes a circuit about the origin (equivalently, a circuit about infinity), the change in the argument of $f$ is precisely $2\pi$, and the function remains unchanged. We have thus continuously traversed an arbitrarily small circuit about infinity, and hence infinity is not a branch point of $f$.
As a result, we only need to choose a branch cut which includes $1$ and $-1$. The real inverval $[-1,1]$ is the simplest choice.