I am still confused. Let me rephrase the question again.

In the textbook, the definition of "arg" and "Arg" are:

$

arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta

$

which means $arg(z) \in \mathbb{R}$

while

$

Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in [-\pi, \pi)

$

which means $Arg(z) \in [-\pi, \pi)$

While in the lecture, as you have introduced, it is the opposite and the range changes to $(-\pi, \pi]$ instead of $[-\pi, \pi)$ (unless I remember incorrectly):

Arg is defined to be

$

Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = (cos\theta + isin\theta)

$

which means $arg(z) \in \mathbb{R}$

while arg is

$

arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in (-\pi, \pi]

$

I am confused because if I am using the definition by the book,

when $z \in \{z : Re (z) > 0\}$

then $arg(z) \in (-\frac{\pi}{2} + 2\pi n,\frac{\pi}{2} + 2\pi n), n \in \mathbb{Z}$