Da stößt der `pgf`-Matheparser an seine Grenzen. Mit `\fpeval` geht's noch.
\documentclass{article}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{xfp}
\usetikzlibrary{plotmarks}
\begin{document}
\begin{tikzpicture}[
x=3cm, y=3cm, scale=0.75,
font=\footnotesize,
>=latex,
]
% Gitternetzlinien
\draw[help lines] (-1.5,-1.5) grid (1.5,1.5);
% x-Achse
\draw[->] (-1.5,0) -- (1.5,0) node[below] {Re};
%Zahlen auf x-Achse
\foreach \x in {-1,1,}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {$\x$};
% y-Achse
\draw[->] (0,-1.5) -- (0,1.5) node[left] {Im};%node[above left]
%Zahlen auf y-Achse
\foreach \y in {-1,1}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {$\y$};
%Ursprung
\node[below left] {$0$};
%%%%%%%%%%%%%%%%
%Folge/Dotplot%%
%%%%%%%%%%%%%%%%
\def\nMax{2000}
\def\nMin{-\nMax}
\foreach \n in {\nMin,...,\nMax}{%====================
\draw[->,thin] plot[mark=*, mark options={color=red, fill=white}, mark size=1.75pt]
coordinates{
( \fpeval{cos((4*\n+1)/2)}, \fpeval{sin((4*\n+1)/2)} )
( \fpeval{cos((4*(\n+1)+1)/2)}, \fpeval{sin((4*(\n+1)+1)/2)} )
};
}%====================
%Titel:
\node[red!95!black, align=left] at (1,1.2) {$
\begin{aligned}
&z_k = e^{\frac{4k+1}{2}i} \\
& k = \nMin,\dots, \nMax
\end{aligned}
$};
\end{tikzpicture}
\end{document}
Braucht allerdings ganz schön lang.
<pre><code>real 0m23.166s
user 0m23.156s
sys 0m0.008s
</code></pre>
Wenn du nicht ganz so lang warten möchtest, dann nimm statt `\usepackage{xfp}`
\directlua{
cos = math.cos
sin = math.sin
}
\def\fpeval#1{\directlua{tex.sprint(#1)}}
<pre><code>real 0m16.415s
user 0m16.324s
sys 0m0.048s
</code></pre>
![alt text][1]
[1]: http://texwelt.de/wissen/upfiles/test_349.png
