About the definition of Argument (in book)

[Ende Jin]

I found that the definition of "arg" and "Arg" in the book is different from that introduced in the lecture (exactly opposite) (on page 7).
I remember in the lecture, the "arg" is the one always lies in $(-\pi, \pi]$
Which one should I use?

[Victor Ivrii]

Which one should I use?

This is a good and tricky question because the answer is nuanced:
$\renewcommand{\Re}{\operatorname{Re}}\renewcommand{\Im}{\operatorname{Im}}$
Solving problems, use definition as in the Textbook, unless the problem under consideration requires modification: for example, if we are restricted to the right half-plane  $\{z\colon \Re z >0\}$ then it is reasonable to consider $\arg z\in (-\pi/2,\pi/2)$, but if we are restricted to the upper half-plane  $\{z\colon \Im z >0\}$ then it is reasonable to consider $\arg z\in (0,\pi)$ and so on.

[Ende Jin]

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

[Victor Ivrii]

$\newcommand{\Arg}{\operatorname{Arg}}\newcommand{\Ln}{\operatorname{Ln}}$ I looked through several popular textbooks and they seem to be equally divided in each issue. I will try to follow the Textbook in $\arg (z)$ and $\Arg (z)$ (and then $\ln (z)$ and $\Ln (z)$). If I follow my habit, you remind me instantly during the lecture.

BTW, you need to write \sin t and \cos t and so on to have them displayed properly (upright and with a space after): $\sin t$, $\cos t$ and so on

[Ende Jin]

Thus in a test/quiz/exam, I should follow the convention of the textbook, right?

[Victor Ivrii]

Thus in a test/quiz/exam, I should follow the convention of the textbook, right?

Indeed

[oighea]

The $\arg$ of a complex number $z$ is an angle $\theta$. All angles $\theta$ have an infinite number of "equivalent" angles, namely $\theta =2k\pi$ for any integer $k$.

Equivalent angles can be characterized by that they exactly overlap when graphed on a graph paper, relative to the $0^\circ$ mark (usually the positive $x$-axis). Or more mathematically, they have the same sine and cosine. It also makes sine and cosine a non-reversible function, as given a sine or cosine, there are an infinite number of angles that satisfy this property.

$\Arg$, on the other hand, reduces the range of the possible angles such that it always lie between $0$ (inclusive) to $2\pi$ (exclusive). That is because one revolution is $2\pi$, or $360$ degrees. That is called the principal argument of a complex number.

We will later discover that complex logarithm also have a similar phenomenon.

[Victor Ivrii]

oighea

Please fix your screen name and use LaTeX command (rendered by MathJax) to display math, not paltry html commands.
I fixed it in this post