# How is the principal branch of logarithm defined?

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 $\operatorname{Im log} z = \pi$.

• you are right, $\log$ takes $\pi$ as an imaginary part on negative reals, this definition although may exclude negative reals (as a line where $\log$ is not continuous) – Kamil Jarosz Mar 16 '16 at 15:33
• At $\pi$ you have what's called a "branch cut" I am with you on the definition in your textbook. That's how I learned it too. It becomes important when you, for example, take $ln(-1)$ which becomes $i\pi$ so the inclusion of $\pi$ is important. BTW, the TI-83 also returns that answer. – imranfat Mar 16 '16 at 15:39

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