Branch cut and principal value

Question

I do not understand the principal value and it is relation to branch cut.
Please tell me about principal value with some examples, then explain the branch cut concept.
For instance, what is the $\text{Arg} (-1-i)$ , tell me your thinking steps.
Thanks

Answer 1

In Complex Analysis, we usually have defined $\arg(z)$ and $\text{Arg}(z)$ where the later generally denotes the principal argument. Most books that I have dealt with define the principal argument to lie in $(-\pi, \pi)$ but it is not unheard of to see it defined between $(0, 2\pi)$.

If we consider $z = -1 - i$, then we have \begin{align} \arg(z) &= \arctan(1)\\ &= \frac{5\pi}{4} + 2\pi k\\ \text{Arg}(z) &= \frac{-3\pi}{4} \end{align} We have to remember to mind the quadrant that the point $z$ lies in when taking the $\arctan$

As I said in the comments, Wikipedia has a good explanation of the branch cut

"A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always, taken between pairs of branch points.(Wikipedia)"