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编写双列证明的简洁方法

tag-icon • tag-icon two-column minipage semantics
编写双列证明的简洁方法

我想按照以下格式写证明。左栏用于练习和解决方案。右栏中我写下我在证明中使用的结果。例如：

在此处输入图片描述

我可以这样做，但我想知道我是否可以写得更简洁一些。理想情况下，我希望有类似以下内容的内容：

\begin{exercise}
    \begin{question}
    The exercise. [in the left-column]
    \end{question}
    \begin{proof}
    The proof. [also in the left-column]
    \end{proof}
    \begin{results}
    Some used results. [in the right-column]
    \end{results}
\end{exercise}

此时图像的代码如下：

\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{framed}
\theoremstyle{definition}
\newtheorem{ex}{Exercise}
\usepackage[top=0.4in, bottom=0.4in, left=0.4in, right=0.4in]{geometry}

%preserves parkskip in the minipage
\usepackage{parskip}
\setlength{\parskip}{\medskipamount}
\makeatletter
\newcommand{\@minipagerestore}{\setlength{\parskip}{\medskipamount}}
\makeatother

\begin{document}
\begin{minipage}[t]{0.55\textwidth}
    \begin{ex}
        Show that \(e^ze^w=e^{z+w}\) for all \(z,w\in\mathbb{C}\).
    \end{ex}

    \begin{proof}
    Let \(z,w\in \mathbb{C}\). We now have that
        \begin{align*}
        e^ze^w&=\sum _{n=0}^\infty \frac{z^n}{n!}\sum _{n=0}^\infty \frac{w^n}{n!}
        \\&=\sum _{n=0}^\infty \bigg(\sum _{k=0}^n \frac{z^k}{k!}\cdot \frac{w^{n-k}}{(n-k)!}\bigg)
        \\& = \sum _{n=0}^\infty  \frac{1}{n!} \bigg(\sum _{k=0}^n {n\choose k} z^k w^{n-k}\bigg)
        \\&=\sum _{n=0}^\infty  \frac{(z+w)^n}{n!}
        \\&= e^{z+w}.
        \end{align*}
    \end{proof}
\end{minipage}
\hfill
\begin{minipage}[t]{0.4\textwidth}
    {\abovedisplayskip=0pt\abovedisplayshortskip=0pt\belowdisplayskip=0pt\belowdisplayshortskip=0pt

    \begin{framed}
        \begin{equation*}
        \sum _{n=0}^\infty a_{n} \sum _{n=0}^\infty b_{n}=\sum _{n=0}^\infty \sum _{k=0}^na_{k}b_{n-k}
        \end{equation*}
    \end{framed}

    \begin{framed}
        \begin{equation*}
        (x+y)^n=\sum _{k=0}^n {n\choose k}x^k y^{n-k}
        \end{equation*}
    \end{framed}

    \begin{framed}
        \begin{equation*}
        {n\choose k}=\frac{n!}{k!\cdot (n-k)!}
        \end{equation*}
    \end{framed}

    \begin{framed}
        \begin{equation*}
        e^z=\sum _{n=0}^\infty  \frac{z^n}{n!} \qquad \forall z \in \mathbb{C}
        \end{equation*}
    \end{framed}
    }
\end{minipage}

答案1

您可以尝试使用paracol包。它在制作并行多列文本方面做得相当不错

http://www.ctan.org/tex-archive/macros/latex/contrib/paracol

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