在对齐环境内，拆分方括号内的两行方程

Question

我有以下等式：

\begin{align}
\left[
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n
\end{align}

我想通过使用\\例如来拆分它，但是出现错误：

\begin{align}
\left[
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +
\\
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n
\end{align}

---

编辑

\begin{align}
...
&= \Bigg\{ \Bigg\{ 1 + m \, \frac{t}{\sigma \, \sqrt{n}} + \mathbb{E} [X_1^2] \, \frac{1}{2} \left(\frac{t}{\sigma \, \sqrt{n}}\right)^2 + O\left[ \left( \frac{t}{\sigma \, \sqrt{n}} \right)^3 \right] \Bigg\}
\\
&\Bigg\{ 1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \left( \frac{m \, t}{\sigma \, \sqrt{n}} \right)^2 + O\left[ \left( \frac{m \, t}{\sigma \, \sqrt{n}} \right)^3 \right] \Bigg\} \Bigg\}^n = \\
&=
\left[
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n 
= \\
&= \left[
1 - \cancel{\frac{m \, t}{\sigma \, \sqrt{n}}} +
\frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\cancel{\frac{m \, t}{\sigma \, \sqrt{n}}} -
\frac{m^2 \, t^2}{\sigma^2 \, n} +
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n = \\
&=
\left[
1 - \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} +
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n = \\
&=
\left[
1 +
\left( \mathbb{E} [X_1^2] - m^2 \right)
\, \frac{t^2}{2 \, \sigma^2 \, n} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n = 
&& \text{(g)} \\
&=
\left[
1 + \sigma^2 \frac{t^2}{2 \, \sigma^2 \, n} + O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n = \\
&=
\left[ 1 + \frac{t^2}{2 \, n} + O \left( \frac{t^3}{n^{3/2}} \right) \right]^n
\end{align}

Answer 1

像这样？

对于这个align环境来说，这不是正确的选择。相反，你应该alined在某些数学环境中使用它（例如equation或\[ ... \]），或者像我在以下 MWE 中所做的那样：使用包multiline中定义的环境mathtools。

\documentclass{article}
\usepackage{amssymb, mathtools}


\begin{document}
    \[
\left[
    \begin{multlined}
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} +
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +
\\
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
    \end{multlined}
\right]^n
    \]
\end{document}

附录 关于您的评论：如果您出于某种（未知的）原因坚持使用align环境，则您无法在整个方程组周围获得括号，而只能在第一行的开始和第二行的结束处获得括号：

\documentclass{article}
\usepackage{amssymb, amsmath}

\begin{document}
    \begin{align*}
&   \left[  
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} +
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} + \right.
\\
&\ \left.\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
    \right]^n
    \end{align*}
\end{document}

据我理解，你的问题在于第一种可能性。

Answer 2

您可以\left[用\right.和\right]来平衡\left.，但不能保证括号的大小相同。

\documentclass{article}
\usepackage{amsmath, amssymb}

\begin{document}

\begin{multline}
\left[
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +
\right.
\\
\left.
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)
\right]^n
\end{multline}

\end{document}

---

这用于\vphantom强制括号大小相同。保存框只是为了避免输入两次代码。

\documentclass{article}
\usepackage{amsmath, amssymb}

\newsavebox{\leftbox}
\newsavebox{\rightbox}

\begin{document}

\savebox{\leftbox}{$\displaystyle
1 - \frac{m \, t}{\sigma \, \sqrt{n}} + \frac{1}{2} \frac{m^2 \, t^2}{\sigma^2 \, n} + 
\frac{m \, t}{\sigma \, \sqrt{n}} - \frac{m^2 \, t^2}{\sigma^2 \, n} + \frac{1}{2} \frac{m^3 \, t^3}{\sigma^3 \, n^{3/2}} +$}

\savebox{\rightbox}{$\displaystyle
\frac{\mathbb{E} [X_1^2]}{2} \frac{t^2}{\sigma^2 \, n} -
\frac{\mathbb{E} [X_1^2]}{2} \frac{m \, t^3}{\sigma^3 \, n^{3/2}} +
\frac{\mathbb{E} [X_1^2]}{4} \frac{m^2 \, t^4}{\sigma^4 \, n^2} +
O \left( \frac{t^3}{n^{3/2}} \right)$}

\begin{multline}
\left[ \usebox\leftbox \vphantom{\usebox\rightbox} \right.
\\
\left. \usebox\rightbox \vphantom{\usebox\leftbox} \right]^n
\end{multline}

\end{document}