Introduction
Which series expansion does the Complex Argument $\arg\left( z \right) \in \mathbb{C}^{*} = \mathbb{C} \cup \left\{ \hat{\infty} \right\}$ where $z \in \mathbb{C}$ have?
What also interests me: How to derive these formulas?
Clarification
Definition Of $\operatorname{arg}\left( z \right)$
The Complex Argument $\arg\left( \cdot \right)$ has a few definitions. I refer specifically to the definition given by:
$$
\begin{align*}
\arg\left( z \right) &\equiv -\ln\left( \frac{z}{\left| z \right|} \right) \cdot i \tag{1.1}\\
\end{align*}
$$
or
$$
\begin{align*}
z &= \left| z \right| \cdot \exp\left( \arg\left( z \right) \cdot i \right)\\
\end{align*}
$$
For reference see here.
Theoretically, we can simplify the whole thing with a generalization of the Signum Function $\operatorname{sgn}\left( \cdot \right)$ for complex numbers given by $\operatorname{sgn}\left( z \right) \equiv \frac{z}{\left| z \right|}$:
$$
\begin{align*}
\arg\left( z \right) &= -i \cdot \ln\left( \operatorname{sgn}\left( z \right) \right) \tag{1.2}\\
\end{align*}
$$
The relations to the Generalized Arctangent Operator for $2$ Arguments $\operatorname{arctan2}\left( \cdot,\, \cdot \right)$ and the Complex Argument Function given by:
$$
\begin{align*}
\arg\left( z \right) &= \operatorname{arctan2}\left( \Im\left( z \right),\, \Re\left( z \right) \right) + 2 \cdot k \cdot \pi\,\text{where}\, k \in \mathbb{Z}\\
\arg\left( z \right) &= \operatorname{Arg}\left( z \right) + 2 \cdot k \cdot \pi\,\text{where}\, k \in \mathbb{Z}\\
\end{align*}
$$
where $\operatorname{Arg}\left( z \right)$ is the Complex Argument of $z$ that is closest to $0$ and $\operatorname{arctan2}\left( \cdot,\, \cdot \right)$ is given by:
$$
\begin{align*}
\operatorname{arctan2}\left( \Im\left( z \right),\, \Re\left( z \right) \right) = \begin{cases}
2 \cdot \arctan\left(\frac{\Im\left( z \right)}{\sqrt{\left( \Re\left( z \right) \right)^2 + \left( \Im\left( z \right) \right)^{2}} + \Re\left( z \right)}\right) &\text{if}\, \Re\left( z \right) > 0,\\
2 \cdot \arctan\left(\frac{\sqrt{\left( \Re\left( z \right) \right)^2 + \left( \Im\left( z \right) \right)^{2}} – \Re\left( z \right)}{\Im\left( z \right)}\right) &\text{if}\, \Re\left( z \right) \leq 0 \text{and}\, \Im\left( z \right) \neq 0,\\
\pi &\text{if}\, \Re\left( z \right) < 0 \text{and}\, \Im\left( z \right) = 0,\\
\text{undefined or}\, \hat{\infty} &\text{if}\ \Re\left( z \right) = 0\, \text{and}\, \Im\left( z \right) = 0.\\
\end{cases} \tag{2}
\end{align*}
$$
where $\Re$ is the Real Part and $\Im$ is the Imaginary Part.
This gives us the plot (where $\arg\left( z \right) \in \left[ -\pi,\, \pi \right]$ and $\Re\left( z \right) \in \left( -10,\, 10 \right) \ni \Im\left( z \right)$:
Question
I'm wondering if $\arg\left( z \right)$ has a series expansion and what is it given by?
It doesn't matter what kind of series expansion, as long as it works.
You might be wondering why I'm asking this: I'm currently working on generalizations of $\arg\left( \cdot \right)$, and I wanted to make a generalization via series expansion, but I didn't know of any for $\arg\left( \cdot \right)$. I then wanted to look up the series development, but my sources such as Wolfram|Functions, MathWorld, Wikipedia and Literature (My Books) did not give me an answer to my search. Now I'm just wondering what series expansions $\arg\left( \cdot \right)$ has?
My Thoughts
Taylor Series
My first idea would be the Taylor Series expansion, but there is a problem: the definitions of the derivative of $\arg\left( \cdot \right)$ are somewhat diverse.
For example, we can see $\arg\left( \cdot \right)$ as a generalization from the Heaviside Step Function $\theta\left( \cdot \right)$ ($\pi \cdot \theta\left( -x \right) = \arg\left( x \right)$ where $x \in \mathbb{R} \backslash \left\{ 0 \right\}$) and thus write the derivative in terms of generalization of some kind of the Dirac Delta Functions $\delta\left( \cdot \right)$.
Laurent-Series
My second idea would be a Laurent Series since $\arg\left( z \right)$ has singulrity at $z = 0$, but I'm not able to compute that.
Mercator Series
We could also use the Mercator Series for the formulas $1.1$ and $1.2$, which is given by $\ln\left( z \right) = \sum\limits_{k = 1}^{\infty}\left[ \frac{\left( -1 \right)^{k + 1}}{k} \cdot \left( z – 1 \right)^{k} \right]$ (a Taylor Series from $\ln$ around $z = 1$):
$$
\begin{align*}
\arg\left( z \right) &\equiv -i \cdot \ln\left( \frac{z}{\left| z \right|} \right) = \sum\limits_{k = 1}^{\infty}\left[ \frac{\left( -1 \right)^{k + 1}}{k} \cdot \left( \frac{z}{\left| z \right|} – 1 \right)^{k} \right]\\
\arg\left( z \right) &\equiv -i \cdot \ln\left( \operatorname{sgn}\left( z \right) \right) = \sum\limits_{k = 1}^{\infty}\left[ \frac{\left( -1 \right)^{k + 1}}{k} \cdot \left( \operatorname{sgn}\left( z \right) – 1 \right)^{k} \right]\\
\end{align*}
$$
So we get:
$$
\fbox{$\begin{align*}
\arg\left( z \right) &= \sum\limits_{k = 1}^{\infty}\left[ \frac{\left( -1 \right)^{k + 1}}{k} \cdot \left( \frac{z}{\left| z \right|} – 1 \right)^{k} \right] + 2 \cdot k \cdot \pi\\
\arg\left( z \right) &= \sum\limits_{k = 1}^{\infty}\left[ \frac{\left( -1 \right)^{k + 1}}{k} \cdot \left( \operatorname{sgn}\left( z \right) – 1 \right)^{k} \right] + 2 \cdot k \cdot \pi\\
\end{align*}$} \tag{3}
$$
But the series only converges for a few values, which doesn't really make it what I'm looking for. More precisely, this series is non-convergent for almost all values of interest.
Series Expansion Of Iverse Trigonometric Functions (incl. $\operatorname{arctan2}$)
Here we come to the same problem as with the logarithm.
Best Answer
You could expand the Taylor series for $\ln(z)$ at other points (e.g. $z = \pm i$) to extend the values of $z$ for which the series converges. Because the argument of $z$ is the same as that of $z/|z|$ (on the unit circle), these 3 patches of $\ln(z)$ will cover about 3/4 of the unit circle. One can extend further, by considering 2 different Taylor series of $\ln(z)$ at $z=-1$; one using $\ln(−1)=−i\pi$ and one using $\ln(−1)=i\pi$ as the constant term of the series (the higher derivative terms will be the same). In this way we get full coverage of the unit circle.
Further Remarks: the above was mostly a copy-paste of my previous comment, but since you are so kind to reward the bounty to me, I felt I should provide a few more "insights". I saw you suggest Laurent series; this can't work here because Laurent series only works when the singularity is isolate, whereas the singularity here (in the argument and logarithm functions) are on a whole slit. Also, you are right to see something deep/interesting about the argument function. It is simultaneously a bit elementary (everyone learns about angles very early on!), but also very deep (the best kind of mathematics!). In particular, its multivalued nature (and the "monodromy" problem of putting together patches of the logarithm function like I was suggesting above) lead to incredible riches:
For these reasons I would also suggest that it is unlikely that there are any generalizations of the argument function to other number systems that would enjoy similar properties of the 2-dimensional argument; this is because it is deeply intertwined with a lot of 2-dimensional topology/geometry, which is somehow very very special (I myself have asked (a) question(s) on MSE about why the number 2 is so special in math, particularly geometry...).
My favorite notes on complex analysis are Terry Tao's blog notes. I wish you encouragement in your mathematical journey! Keep asking questions, even if they seem a bit off the beaten track; sometimes profound insights about the beaten track (both the parts laid already, and the parts yet to be laid) can be found off of it (but of course, at the same time, read up on the beaten track; it's beaten for a reason!).