Natural domain to define f(z)=loglog

[justin_huang]

Homework Statement

Let log z be the principal branch of the logarithm function defined −pi < arg z < pi
the domain to define f(z)=loglog z??

Homework Equations

The Attempt at a Solution

I already know how to represent the arg z by arctanb/a+2pi*k
how can I get arctan [Re (log z)/Im (log z)]+2pi*k
or any other method?

 

[micromass]

I already know how to represent the arg z by arctanb/a+2pi*k

The "$+2\pi k$ shouldn't be here right?

how can I get arctan [Re (log z)/Im (log z)]+2pi*k
or any other method?

So basically, you want to know when

$$-\pi<arg(log(z))<\pi$$

surely this correspond to log(z) being a negative real number.

So, you must calculate log(z) in some way (do you have a formula for it) and see when it is a negative real number. These are the z we don't want.

 

[micromass]

I considered log(z)=0 to be part of "log(z) is a negative real number", but this is probably not standard terminology. So yes, you also need to make sure that log(z) isn't 0...

 

[justin_huang]

The "$+2\pi k$ shouldn't be here right?



So basically, you want to know when

$$-\pi<arg(log(z))<\pi$$

surely this correspond to log(z) being a negative real number.

So, you must calculate log(z) in some way (do you have a formula for it) and see when it is a negative real number. These are the z we don't want.

thanks so much