- #1

fatpotato

- Homework Statement
- Using the definition of a branch to show that ##Arg z## is a branch of ##\arg z##

- Relevant Equations
- Definition of a branch, definition of principal argument ##Arg z##, definition of argument function ##\arg z##

I find the following definition in my complex analysis book :

Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one of the values of ##f(z)##.

I am now assuming that the choosen domain is in fact ##\mathbb{C}##, an open and connected set. Now, the first function given as an example for this definition is ##Arg z##, the principal argument of ##z## (is there a latex command for this function?) that relates to ##\arg z## with ##Arg z = \arg_{-\pi} z \in (-\pi;\pi]## where ##Arg z## is simply the argument of the complex number ##z \neq 0## that can take any value in the interval ##(-\pi;\pi]##.

Acording to the book, ##Arg z## is a branch of ##\arg z## because (considering ##z \neq 0##):

Anyone has an idea?

Edit: missing sentence, add condition ##z\neq 0##

Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one of the values of ##f(z)##.

I am now assuming that the choosen domain is in fact ##\mathbb{C}##, an open and connected set. Now, the first function given as an example for this definition is ##Arg z##, the principal argument of ##z## (is there a latex command for this function?) that relates to ##\arg z## with ##Arg z = \arg_{-\pi} z \in (-\pi;\pi]## where ##Arg z## is simply the argument of the complex number ##z \neq 0## that can take any value in the interval ##(-\pi;\pi]##.

Acording to the book, ##Arg z## is a branch of ##\arg z## because (considering ##z \neq 0##):

- It is single-valued on ##\mathbb{C}## - I'm ok with this
- ##Arg z## is one of the values of ##\arg z## - I'm still ok with this
- It is continuous on ##\mathbb{C}## - I'm not okay with this!

*branch cut*on the negative real line, there will be a jump from ##-\pi## to ##\pi##, will it not? The book even makes a point of insisting that there will always be a jump of ##2\pi## when considering the argument of a complex number. This confuses me greatly, and I need branches in the next chapter since they are used to define the behaviour of the complex logarithm.Anyone has an idea?

Edit: missing sentence, add condition ##z\neq 0##