I Merging Two Threads: Complex Integrals & Branch Cuts

[Silviu]

<Moderator note: Merger of two threads on the topic.>

Hello! I am reading some basic stuff on complex integrals using branch cuts and i found the problem in the attachment. I am not sure I understand why the branch cut is along $R^+$. I thought that branch cut is, loosely speaking, a line where the function is not continuous (and thus not holomorphic). But in the presented problem, the function is continuous on $R^+$ as $lim_{\theta \to 0} = \sqrt{r}$ and $lim_{\theta \to 2\pi} = -\sqrt{r}$. The limits are not equal, but they don't have to be, as the funtion is not defined for $\theta = 2\pi$. However, the function is not continuous for $\theta = \pi$, as, coming from above and below x-axis, gives different values for $sin(\theta)$. So, isn't the branch cut on $R^-$, or did I get something wrong about the definition of branch cut? Thank you!

 

[Silviu]

Hello! I understand that the branch cuts are meant to prevent a complex function from being multivalued. So, as any complex number $z$ can be written in different ways ($z=\|z\|e^{i\theta}=\|z\|e^{i(\theta +2\pi)}$ etc.), does this mean that any complex function has a branch cut, depending on the interval on which we define $\theta$, ($[0,2\pi)$ or $[-\pi,\pi)$, etc)?

 

[arpon]

Look, functions like $f(z)=z^2$, gives you the same value for a particular $z$, no matter you write $z=||z||e^{i\theta}$ or $z=||z||e^{i(\theta+2\pi)}$.
The problem arises when you deal with functions like $g(z) = log (z)$ or $g(z) = z^{1/2}$. In those cases, you notice, $z=||z||e^{i\theta}$ or $z=||z||e^{i(\theta+2\pi)}$ give different values of $g(z)$.

 

[mathwonk]

the fact that theta is multivalued is irrelevant to the function e^(i.theta) since the periodicity of the exponential function cancels out the multivaluedness of theta.

 

[WWGD]

Another perspective is that of branch points . You want a branch cut designed so that curves do not wind around the branch point. EDIT: For example, for logz, the Complex log, zero is the branch point and branch cuts disallow winding around it.

 

[Silviu]

But how do you calculate a branch cut in general? For example $f(z)=\sqrt{z(z-1)}$, how do I get branch points and how do I get the number of values you get while going around the branch points? Thank you!