In my textbook, it is defined as:
$$\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$$
Where $\operatorname{Arg}$ is the principal branch of $\arg$, that's, the function which outputs the unique argument of $z$ in the interval $(-\pi, \pi]$.
However, I am reading from Ahlfors' Complex Analysis, and on page $71$, it is written: "…define the principal branch of logarithm by the condition $| \operatorname{Im log} z | < \pi$".
Which one is the correct definition? They are essentially different definitions because one includes the possibility that $\operatorname{Im log} z = \pi$.
Best Answer
Conventions differ, and there is not necessarily a single "right" way. Ahlfors is reluctant to give a function a value on its branch cut, because that makes the function discontinuous. He would prefer to talk about limits as $z$ approaches a point on the branch cut from one side or the other. More "applied" treatments may find it convenient to specify an actual value for the function on the branch cut. You want your calculator to give you an answer when you enter