Simple confirmation about the main branch of the complex Logarithm.

Question

As far as I know, we define the main branch of the complex Logarithm by fixing the argument beetween $(-\pi,\pi]$ (The closed Interval could be in one of the both sides). And so we have that: \begin{equation*} Log(z) = ln|z|+iArg(z), \quad Arg(z) \in (-\pi,\pi] \end{equation*}

For $z=1:$ \begin{equation*} Log(1) = ln|1|+i0 = ln|1| = ln(1) = 0 \end{equation*} So my question is the following:

Can we define the main branch just by saying that it is the branch that makes $log(1)=0$ ? Or should we always refer to the fact the we fix the argument beetween the refered interval?

Thanks for all the help in advance.

Answer 1

You need to specify the interval on which you define the argument or your function isn’t well-defined.

For example, say we only specify that $\log(1)=0$. Then what is $\log(-1)$? Well $e^{i\pi}=-1$ and $e^{-i\pi}=-1$ so we could feasibly give two values to $\log(-1)$, which means that we no longer have a well-defined function.