I do not understand this excerpt from my textbook:

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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.

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    $\begingroup$ 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. $\endgroup$ – 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.

The set given in Exercise IV.10.1 is not a open disk -- it's annulus (one that happens to "enclose" the origin).

I think perhaps your confusion might be that you're reading "open disk, with the origin removed" for "open disk that excludes the origin," but I may be misunderstanding.

Regardless, the claim is that on certain open disks --- those in which the origin is not one of the points --- the branch is defined. The set in the exercise is not an open disk, so there's no contradiction.

  • $\begingroup$ Thank you, you are right, I was reading it as 'disk centered at {0} but not including {0}.' Just to be clear, does this mean (1) this branch property never holds for an open disk that includes the origin, and that (2) for every open disk that does not contain {0}, there is a branch of arg z and log z? $\endgroup$ – sfdisk May 14 at 15:46
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    $\begingroup$ Both of those statements are true (assuming "this branch property" means "there's a well-defined function "arg" with the property that ..."). I'm not sure that I can say that "this means that...", because I don't know what "this" is, precisely. But both statements are indeed true. $\endgroup$ – John Hughes May 14 at 16:23
  • $\begingroup$ 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 ..." $\endgroup$ – Fareed AF May 22 at 8:31
  • $\begingroup$ I have no idea what those "deprived of" clauses mean. Perhaps the book is a translation, and these are translation errors? $\endgroup$ – John Hughes May 22 at 10:04

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