Branches of $\arg(z)$ and $\log(z)$

I do not understand this excerpt from my textbook: It says that any open disk G that excludes the origin, G has a branch. But in the problem, it says there is no such branch? I do not understand the wording.

• I agree that the wording is misleading. I think it means that any point within the inner ring of a disc is also "within" the disc. – John Doe May 14 at 15:33

An "open disk that excludes the origin" is something like $$\{z : |z-2| < 1 \}$$ -- and open disk centered at $$2$$, with radius $$1$$. The point $$z = 0$$ is not in this set. If I'd written $$\{z : |z-2| < 3 \}$$ it would be an open disk that DOES include the origin.
• I have a small question please, I read on a copybook about complex numbers, a sentence that said "if $z$ is a real number then $$arg z= 0, \pi$$ deprived of the denominator". And a similar one "if $z$ is pure immaginary then $$arg z= \pm \pi/2$$, deprived of the numerator and denominator". Any idea what could this possibly mean? Thanks in advance ... By the way I am just asking about the sentence "deprived of ..." – Fareed AF May 22 at 8:31