如何绘制复数的 5 次方根

Question

我想画出 的根z^5=1-i。

我对 Latex 完全是新手，对绘画真的一无所知。谢谢你的帮助！

Answer 1

嗯，似乎没人想回答这个问题。我同意这个问题与链接的问题非常相似，但个人认为这不是重复问题，因为你要求的不是求 n 次方根，而是求具有非平凡半径和相位的不同复数。

\documentclass[fleqn]{article} 
\usepackage{tikz}
\usepackage{amsmath}
\begin{document}
You wish to plot the 5th roots of 
\[ 1-\mathrm{i}~=~\sqrt{2}\,\mathrm{e}^{-\mathrm{i}\pi/4}\]
for some complex number $z=r\cdot\mathrm{e}^{\mathrm{i}\,\varphi}$.
This means that you need to solve the equation
\[ z^5~=~r^5\,\mathrm{e}^{5\cdot\mathrm{i}\cdot\varphi}
~=~\sqrt{2}\,\mathrm{e}^{-\mathrm{i}\pi/4}\;,\]
which translates to 
\[r~=~2^{1/10}\quad\text{and}\quad \varphi~=~\frac{\pi}{20}\left(-1+8\,n\right)
~=~9^\circ\left(-1+8\,n\right)
\]
with $n\in\{0,...,4\}$.

\begin{tikzpicture}[scale=3]
\pgfmathsetmacro{\ticklength}{0.06}
\draw [-latex] (-2,0) -- (2,0) node[below left]{Re$\,z$};
\draw [-latex] (0,-2) -- (0,2) node[below left]{Im$\,z$};
\draw (1,\ticklength) -- (1,-\ticklength) node[below] {1};
\draw (\ticklength,1) -- (-\ticklength,1) node[left] {i};
\draw (0,0) circle({pow(2,1/10)});
\foreach \X in {0,...,4}
{\node[scale=0.4,circle,fill,label={{9*(-1+8*\X)}:$n=\X$}] at ({9*(-1+8*\X)}:{pow(2,1/10)}) {}; }
\end{tikzpicture}

\end{document}