Ref: http://blog.sina.com.cn/s/blog_5e16f1770100fs38.html
下载LaTeX最强大的功能就是显示美丽的数学公式，下面我们来看这些公式是怎么实现的。
1、数学公式的前后要加上 $ 或 \( 和 \)，比如：$f(x) = 3x + 7$ 和 \(f(x) = 3x + 7\) 效果是一样的；
如果用 \[ 和 \]，或者使用 $$ 和 $$，则改公式独占一行；
如果用 \begin{equation} 和 \end{equation}，则公式除了独占一行还会自动被添加序号， 如何公式不想编号则使用 \begin{equation*} 和 \end{equation*}.
1、数学公式的前后要加上 $ 或 \( 和 \)，比如：$f(x) = 3x + 7$ 和 \(f(x) = 3x + 7\) 效果是一样的；
如果用 \[ 和 \]，或者使用 $$ 和 $$，则改公式独占一行；
如果用 \begin{equation} 和 \end{equation}，则公式除了独占一行还会自动被添加序号， 如何公式不想编号则使用 \begin{equation*} 和 \end{equation*}.
2、字符
普通字符在数学公式中含义一样，除了
# $ % & ~ _ ^ \ { }
若要在数学环境中表示这些符号# $ % & _ { }，需要分别表示为\# \$ \% \& \_ \{ \}，即在个字符前加上\。
3、上标和下标
用 ^ 来表示上标，用 _ 来表示下标，看一简单例子：
$$\sum_{i=1}^n a_i=0$$
$$f(x)=x^{x^x}$$
$$f(x)=x^{x^x}$$
效果:
这里有更多的LaTeX上标下标的设置
4、希腊字母
数学公式离不开希腊字母，以下列出了LaTex中产生希腊字母的控制命令：
下面我们就列举我们常用的希腊符号的几种形式
大小写
\Alpha 就是\alpha的大写
正体斜体
代码如下：
\begin{tabular}{|c|ccccccccccc|}
\hline
正体&$\Gamma$ & $\Delta$ & $\Theta$ & $\Lambda$ & $\Xi$ & $\Pi$ & $\Sigma$ & $\Upsilon$ & $\Phi$ & $\Psi$ & $\Omega$\\
\hline
\verb|\mit|斜体&$\mit\Gamma$ & $\mit\Delta$ & $\mit\Theta$ & $\mit\Lambda$ & $\mit\Xi$ & $\mit\Pi$ & $\mit\Sigma$ & $\mit\Upsilon$ & $\mit\Phi$ & $\mit\Psi$ & $\mit\Omega$\\
\hline
\end{tabular}
amsmath.sty,amssymb.sty所提供的符号
代码如下：
\begin{tabular}{|c|ccccccccccc|}
\hline
正体&$\Gamma$ & $\Delta$ & $\Theta$ & $\Lambda$ & $\Xi$ & $\Pi$ & $\Sigma$ & $\Upsilon$ & $\Phi$ & $\Psi$ & $\Omega$\\
\hline
\verb|\mit|斜体&$\mit\Gamma$ & $\mit\Delta$ & $\mit\Theta$ & $\mit\Lambda$ & $\mit\Xi$ & $\mit\Pi$ & $\mit\Sigma$ & $\mit\Upsilon$ & $\mit\Phi$ & $\mit\Psi$ & $\mit\Omega$\\
\hline
\end{tabular}
amsmath.sty,amssymb.sty所提供的符号
5、数学函数
LATEX 中用斜体显示变量名, 用正体显示函数名. 在LATEX 中通过在函数名前加
上倒斜线, 将函数名与变量名区分开. 但这样的函数名需要事先定义, LATEX 中已定
义了以下函数名（即以下函数名可以在 LATEX 中直接使用）:
arccos \arccos coth \coth hom \hom ln \ln sinh \sinh
arcsin \arcsin csc \csc inf \inf log \log sup \sup
arctan \arctan deg \deg ker \ker max \max tan \tan
arg \arg det \det lg \lg min \min tanh \tanh
cos \cos dim \dim lim \lim Pr \Pr mod \bmod
cosh \cos exp \exp lim inf \liminf sec \sec (mod ) \pmod
cot \cot gcd \gcd lim sup \limsup sin \sin
下面是介绍如何使用 \DeclareMathOperator 或 \DeclareMathOperator* 命
令来自定义函数名。
To de¯ne additional named operators outside the above list, use the
\DeclareMathOperator command; for example, after
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\esssup}{ess\,sup}
in preamble, one could write
\rank(x) ¡! rank(x)
\esssup(y,z) ¡! ess sup(y; z)
The star form \DeclareMathOperator* creates an operator that takes limits in a
displayed formula like sup or max.
When prede¯ning such a named operator is problematic (e.g., when using one
in the title or abstract of an article), there is an alternative form that can be used
directly:
\operatorname{rank}(x) ¡! rank(x)
上倒斜线, 将函数名与变量名区分开. 但这样的函数名需要事先定义, LATEX 中已定
义了以下函数名（即以下函数名可以在 LATEX 中直接使用）:
arccos \arccos coth \coth hom \hom ln \ln sinh \sinh
arcsin \arcsin csc \csc inf \inf log \log sup \sup
arctan \arctan deg \deg ker \ker max \max tan \tan
arg \arg det \det lg \lg min \min tanh \tanh
cos \cos dim \dim lim \lim Pr \Pr mod \bmod
cosh \cos exp \exp lim inf \liminf sec \sec (mod ) \pmod
cot \cot gcd \gcd lim sup \limsup sin \sin
下面是介绍如何使用 \DeclareMathOperator 或 \DeclareMathOperator* 命
令来自定义函数名。
To de¯ne additional named operators outside the above list, use the
\DeclareMathOperator command; for example, after
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\esssup}{ess\,sup}
in preamble, one could write
\rank(x) ¡! rank(x)
\esssup(y,z) ¡! ess sup(y; z)
The star form \DeclareMathOperator* creates an operator that takes limits in a
displayed formula like sup or max.
When prede¯ning such a named operator is problematic (e.g., when using one
in the title or abstract of an article), there is an alternative form that can be used
directly:
\operatorname{rank}(x) ¡! rank(x)
6、在公式中插入文本可以通过 \mbox{text} 在公式中添加text，比如：
\documentclass{article}
\usepackage{CJK}
\begin{CJK*}{GBK}{song}
\begin{document}
$$\mbox{对任意的$x>0$}, \mbox{有 }f(x)>0. $$
\end{CJK*}
\end{document}
\usepackage{CJK}
\begin{CJK*}{GBK}{song}
\begin{document}
$$\mbox{对任意的$x>0$}, \mbox{有 }f(x)>0. $$
\end{CJK*}
\end{document}
效果:
7、分数及开方
\frac{numerator}{denominator} \sqrt{expression_r_r_r}表示开平方，
\sqrt[n]{expression_r_r_r} 表示开 n 次方.
8、省略号（3个点）
\ldots 表示跟文本底线对齐的省略号；\cdots 表示跟文本中线对齐的省略号，
比如：
表示为 $$f(x_1,x_x,\ldots,x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 $$
9、括号和分隔符
() 和 [ ] 和 ｜ 对应于自己；
{} 对应于 \{ \}；
|| 对应于 \|。
当要显示大号的括号或分隔符时，要对应用 \left 和 \right，如：
\[f(x,y,z) = 3y^2 z \left( 3 + \frac{7x+5}{1 + y^2} \right).\]对应于
\left. 和 \right. 只用与匹配，本身是不显示的，比如，要输出：
则用 $$\left. \frac{du}{dx} \right|_{x=0}.$$
10、多行的数学公式
可以表示为：
\begin{eqnarray*}
\cos 2\theta & = & \cos^2 \theta - \sin^2 \theta \\
& = & 2 \cos^2 \theta - 1.
\end{eqnarray*}
\cos 2\theta & = & \cos^2 \theta - \sin^2 \theta \\
& = & 2 \cos^2 \theta - 1.
\end{eqnarray*}
其中&是对其点，表示在此对齐。
*使latex不自动显示序号，如果想让latex自动标上序号，则把*去掉
*使latex不自动显示序号，如果想让latex自动标上序号，则把*去掉
11、矩阵
表示为：
The \emph{characteristic polynomial} $\chi(\lambda)$ of the
$3 \times 3$~matrix
\[ \left( \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right)\]
is given by the formula
\[ \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right|.\]
$3 \times 3$~matrix
\[ \left( \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right)\]
is given by the formula
\[ \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right|.\]
c表示向中对齐，l表示向左对齐，r表示向右对齐。
12、导数、极限、求和、积分(Derivatives, Limits, Sums and Integrals)
The expression_r_r_rs
are obtained in LaTeX by typing
\frac{du}{dt} and \frac{d^2 u}{dx^2}
respectively. The mathematical symbol 
is produced using \partial. Thus the Heat Equation
is produced using \partial. Thus the Heat Equation
is obtained in LaTeX by typing
\[ \frac{\partial u}{\partial t}
= h^2 \left( \frac{\partial^2 u}{\partial x^2}
+ \frac{\partial^2 u}{\partial y^2}
+ \frac{\partial^2 u}{\partial z^2}\right)\]
= h^2 \left( \frac{\partial^2 u}{\partial x^2}
+ \frac{\partial^2 u}{\partial y^2}
+ \frac{\partial^2 u}{\partial z^2}\right)\]
To obtain mathematical expression_r_r_rs such as
in displayed equations we type \lim_{x \to +\infty}, \inf_{x > s} and \sup_K respectively. Thus to obtain
(in LaTeX) we type
\[ \lim_{x \to 0} \frac{3x^2 +7x^3}{x^2 +5x^4} = 3.\]
To obtain a summation sign such as
we type \sum_{i=1}^{2n}. Thus
is obtained by typing
\[ \sum_{k=1}^n k^2 = \frac{1}{2} n (n+1).\]
We now discuss how to obtain integrals in mathematical documents. A typical integral is the following:
This is typeset using
\[ \int_a^b f(x)\,dx.\]
The integral sign is typeset using the control sequence \int, and the limits of integration (in this case a and b are treated as a subscript and a superscript on the integral sign.
Most integrals occurring in mathematical documents begin with an integral sign and contain one or more instances of d followed by another (Latin or Greek) letter, as in dx, dy and dt. To obtain the correct appearance one should put extra space before the d, using \,. Thus
Most integrals occurring in mathematical documents begin with an integral sign and contain one or more instances of d followed by another (Latin or Greek) letter, as in dx, dy and dt. To obtain the correct appearance one should put extra space before the d, using \,. Thus
and
are obtained by typing
\[ \int_0^{+\infty} x^n e^{-x} \,dx = n!.\]
\[ \int \cos \theta \,d\theta = \sin \theta.\]
\[ \int_{x^2 + y^2 \leq R^2} f(x,y)\,dx\,dy
= \int_{\theta=0}^{2\pi} \int_{r=0}^R
f(r\cos\theta,r\sin\theta) r\,dr\,d\theta.\]
\[ \int \cos \theta \,d\theta = \sin \theta.\]
\[ \int_{x^2 + y^2 \leq R^2} f(x,y)\,dx\,dy
= \int_{\theta=0}^{2\pi} \int_{r=0}^R
f(r\cos\theta,r\sin\theta) r\,dr\,d\theta.\]
and
\[ \int_0^R \frac{2x\,dx}{1+x^2} = \log(1+R^2).\]
respectively.
In some multiple integrals (i.e., integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. The way to improve the appearance of of the integral is to use the control sequence \! to remove a thin strip of unwanted space. Thus, for example, the multiple integral
is obtained by typing
\[ \int_0^1 \! \int_0^1 x^2 y^2\,dx\,dy.\]
Had we typed
\[ \int_0^1 \int_0^1 x^2 y^2\,dx\,dy.\]
we would have obtained
A particularly noteworthy example comes when we are typesetting a multiple integral such as
Here we use \! three times to obtain suitable spacing between the integral signs. We typeset this integral using
\[ \int \!\!\! \int_D f(x,y)\,dx\,dy.\]
Had we typed
\[ \int \int_D f(x,y)\,dx\,dy.\]
we would have obtained
The following (reasonably complicated) passage exhibits a number of the features which we have been discussing:
One would typeset this in LaTeX by typing In non-relativistic wave mechanics, the wave function
$\psi(\mathbf{r},t)$ of a particle satisfies the
\emph{Schr\"{o}dinger Wave Equation}
\[ i\hbar\frac{\partial \psi}{\partial t}
= \frac{-\hbar^2}{2m} \left(
\frac{\partial^2}{\partial x^2}
+ \frac{\partial^2}{\partial y^2}
+ \frac{\partial^2}{\partial z^2}
\right) \psi + V \psi.\]
It is customary to normalize the wave equation by
demanding that
\[ \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},0) \right|^2\,dx\,dy\,dz = 1.\]
A simple calculation using the Schr\"{o}dinger wave
equation shows that
\[ \frac{d}{dt} \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz = 0,\]
and hence
\[ \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz = 1\]
for all times~$t$. If we normalize the wave function in this
way then, for any (measurable) subset~$V$ of $\textbf{R}^3$
and time~$t$,
\[ \int \!\!\! \int \!\!\! \int_V
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz\]
represents the probability that the particle is to be found
within the region~$V$ at time~$t$.
$\psi(\mathbf{r},t)$ of a particle satisfies the
\emph{Schr\"{o}dinger Wave Equation}
\[ i\hbar\frac{\partial \psi}{\partial t}
= \frac{-\hbar^2}{2m} \left(
\frac{\partial^2}{\partial x^2}
+ \frac{\partial^2}{\partial y^2}
+ \frac{\partial^2}{\partial z^2}
\right) \psi + V \psi.\]
It is customary to normalize the wave equation by
demanding that
\[ \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},0) \right|^2\,dx\,dy\,dz = 1.\]
A simple calculation using the Schr\"{o}dinger wave
equation shows that
\[ \frac{d}{dt} \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz = 0,\]
and hence
\[ \int \!\!\! \int \!\!\! \int_{\textbf{R}^3}
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz = 1\]
for all times~$t$. If we normalize the wave function in this
way then, for any (measurable) subset~$V$ of $\textbf{R}^3$
and time~$t$,
\[ \int \!\!\! \int \!\!\! \int_V
\left| \psi(\mathbf{r},t) \right|^2\,dx\,dy\,dz\]
represents the probability that the particle is to be found
within the region~$V$ at time~$t$.
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