在JupyterNotebook⾥⾯写Python代码和数学公式
这⾥做了更多和语法错误修改。
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 $$ 和 $$，则该公式独占⼀⾏；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
除了# $ % & ~ _ ^ \ { }普通字符在数学公式中含义⼀样，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：
$$\sum_{i=1}^n a_i=0$$
$$f(x)=x^{x^x}$$
效果:
LaTeX可以通过这符号$^$和$_$来设置上标和下标。使⽤可以参见：
⽤ ^ 来表⽰上标，⽤ _ 来表⽰下标，如果上标的内容多于⼀个字符，注意⽤{ }把上标括起来，上下标是可以嵌套的，下⾯是⼀些简单例⼦：
$\sum_{i=1}^n a_i=0$
$f(x)=x^{x^x}$
4、希腊字母
更多请参见
5、数学函数
例如sin x，输⼊应该为\sin 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}
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*}
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
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
\[ \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right|.\]
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
上⾯的公式可以输⼊下⾯代码：
\frac{du}{dt}and \frac{d^2 u}{dx^2}
的代码如下：
\[ \frac{\partial u}{\partial t}
= h^2 \left( \frac{\partial^2 u}{\partial x^2}
+ \frac{\partial^2 u}{\partial y^2}
+ \frac{\partial^2 u}{\partial z^2}\right)\]
为了显⽰以下公式：
代码是 \lim_{x \to +\infty}, \inf_{x > s}and \sup_K respectively.
以下公式
(in LaTeX) 我们输⼊
\[ \lim_{x \to 0} \frac{3x^2 +7x^3}{x^2 +5x^4} = 3.\]
以下公式
的代码如下
\frac{1}{\lim_{u \rightarrow \infty}}, \frac{1}{\lim\limits_{u \rightarrow \infty}} or
\frac{1}{ \displaystyle \lim_{u \rightarrow \infty}} respectively.
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 m
ore 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
\[ \int_0^{+\infty} x^n e^{-x} \,dx = n!.\]
\[ \int \cos \theta \,d\theta = \sin \theta.\]
\[ \int_{x^2 + y^2 \leq R^2} f(x,y)\,dx\,dy
= \int_{\theta=0}^{2\pi} \int_{r=0}^R
f(r\cos\theta,r\sin\theta) r\,dr\,d\theta.\]
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}
\[ i\hbar\frac{\partial \psi}{\partial t}
= \frac{-\hbar^2}{2m} \left(
python新手代码例子
\frac{\partial^2}{\partial x^2}
+ \frac{\partial^2}{\partial y^2}
+ \frac{\partial^2}{\partial z^2}
\right) \psi + V \psi.\]
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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