LaTeX/數學公式

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高納德⋅克魯斯(Donald Knuth,TeX系統的創始人)編寫TeX系統的首要動力之一,就是希望發明一個公式編輯系統,使得用戶可以使用簡單的指令構建專業的數學文檔.事實證明TeX獲得了很大的成功,也因此受到了某些科學團體的青睞.數學排版系統(typesetting mathematics)是LaTeX最主要的優勢之一。並且,由於數學公式和符號十分繁雜,如何使用LaTeX的數學排版系統也是我們的核心話題之一。

如果你的文件只是需要一點點簡單的數學公式,LaTeX中許多工具你是要用到的。如果你寫一篇科技性的文章,它包含大量複雜的公式,這個amsmath 宏包[1] 介紹了許多比LaTeX更加靈活與強大的新命令。 mathtools宏包功能更為強大,保留amsmath所有命令並加入許多十分好用的設定、符號以及環境 [2]。寫作時,將下列命令寫在導言

\usepackage{amsmath}

\usepackage{mathtools}

如果引用mathtools,系統就會自動加載amsmath,故無須再寫\usepackage{amsmath}於導言中。

數學環境(數學環境)[編輯]

LaTeX 需要事先知道後面的文本確實包含數學元素。這是因為 LaTeX 排版數學符號的方式與普通文本不同。因此,特殊的環境為此目的而被聲明。根據它們的呈現方式,它們可以分為兩類:

  • text - 內聯公式(行內公式), 即在聲明文本的正文之內. 例如, 我可以在這句話中表達 .
  • displayed - 外聯公式(行間公式), 即與正文分開的公式.例如, 要在行間顯示一個公式:
  • 行內公式: 在 LaTeX 中, 使用 $ ... $ 來聲明一個行內公式, 或者使用 \( ... \). 被括起來的地方會被解釋為一個行內公式數學環境.
  • 行間公式: 使用 $$ ... $$ 來聲明一個行間公式, 或者使用 \[ ... \].

除此之外, 還可以使用 LaTeX 的環境聲明標籤來聲明數學環境. 以上介紹的部分其實是下面語法的簡寫.

  • 行內公式

\begin{math}

...

\end{math}

  • 行間公式

\begin{displaymath}

...

\end{displaymath}

  • 自動編號的行間公式

\begin{equation}

...

\end{equation}

注意: 應該避免使用 $$ ... $$ , 因為它可能導致一些問題, 尤其是當你使用 AMS-LaTeX 宏包的時候. 此外, 如果出現問題, 錯誤提示可能沒有幫助.(2018年6月26日注:此條可能已不正確,本條內容上次修改時間為2017年1月.)

如果你正常輸入文本, 則你處於 文本模式, 而當你在數學環境中輸入時, 情況會和 文本模式有所不同.

  1. 所有空格與換行都會失去作用. 要在數學模式里使用空格, 你應用 \quad 輸入一個標準寬度的空格. 或者使用 \ 或 \; 或 \, 等輸入寬度不同的空格.
  2. 不允許出現空行, 只能是一個段落一段公式.
  3. 所有字母都會按變量(var)格式排版, 如果需要排版普通拉丁文樣式的字母, 請使用 \mathrm{}. 例如, \mathrm{d} 常用來表現 微分算子

在段落文字內插入並以 "displayed" 樣式顯示公式[編輯]

對於一些大型運算符, 行內與行間樣式會有所不同. 行內公式會壓縮高度:

行內的累加運算符:

行間的累加運算符:

符號[編輯]

數學中存在很多很多的符號!記憶住如何使用LaTex產生它們是很困難的。有一部分符號可以通過鍵盤直接輸入。 Mathematics has lots and lots of symbols! If there is one aspect of maths that is difficult in LaTeX it is trying to remember how to produce them. There are of course a set of symbols that can be accessed directly from the keyboard:

+ - = ! / ( ) [ ] < > | ' : *

Beyond those listed above, distinct commands must be issued in order to display the desired symbols. There are many examples such as Greek letters, set and relations symbols, arrows, binary operators, etc.

For example:

θ


Fortunately, there's a tool that can greatly simplify the search for the command for a specific symbol. Look for "Detexify" in the external links section below. Another option would be to look in "The Comprehensive LaTeX Symbol List" in the external links section below.

希臘字母[編輯]

希臘字母在數學中經常用到,因為他們在數學的模型中很容易輸入。你只需在符號"\"(無引號)後打他們的名字即可得到他們: 若首字母小寫,你將得到小寫的希臘字母,若首字母大寫 (且只有首字母大寫), 你將得到大寫的希臘字母. 注意:一些大寫的希臘字母像拉丁字母, 所以LaTeX將不提供他們 (e.g. uppercase Alpha and Beta are just "A" and "B" respectively). Lowercase epsilon, theta, phi, pi, rho, and sigma are provided in two different versions. The alternate, or variant, version is created by adding "var" before the name of the letter:

θ

Scroll down to #List_of_Mathematical_Symbols for a complete list of Greek symbols.

算子[編輯]

算子是一個用寫成一個單詞的函數: e.g. trigonometric functions (sin, cos, tan), logarithms and exponentials (log, exp). LaTeX has many of these defined as commands: θ

For certain operators such as limits, the subscript is placed underneath the operator: θ

For the modular operator there are two commands: ± and ±: θ θ

To use operators which are not pre-defined, such as argmax, see custom operators

指數和次方數[編輯]

指數與指標(在右上或右下角的小字)就如同一般文字編輯時所說的上標和下標。同樣的,我們為上標加上電腦常用的上標符號^前綴,並且用底線符號_代表下標。注意,如果要寫成上下標的文字或符號超過一個,要用花括號({})將所有要設為上下標的文字框起來。

例如:

  • k_{n+1} = n^2 + k_n^2 - k_{n-1}編譯為
  • f(n) = n^5 + 4n^2 + 2 |_{n=17}編譯為


Powers and indices are equivalent to superscripts and subscripts in normal text mode. The caret (^) character is used to raise something, and the underscore (_) is for lowering. If more than one expression is raised or lowered, they should be grouped using curly braces ({ and }). θ

An underscore (_) can be used with a vertical bar () to denote evaluation using subscript notation in mathematics:

θ

分數及二項式[編輯]

A fraction is created using the ±} command. (for those who need their memories refreshed, that's the top and bottom respectively!). Likewise, the binomial coefficient (aka the Choose function) may be written using the ± command[3]: θ

It is also possible to use the ± command without the amsmath package: θ

You can embed fractions within fractions: θ{y-z} \] |render= }}

Note that when appearing inside another fraction, or in inline text , a fraction is noticeably smaller than in displayed mathematics. The ± and ± commands[3] force the use of the respective styles, ± and ±. Similarly, the ± and ± commands typeset the binomial coefficient.

Another way to write fractions is to use the ± command without the amsmath package: θ

For relatively simple fractions, it may be more aesthetically pleasing to use powers and indices: θ

If you use them throughout the document, usage of xfrac package is recommended. This package provides ± command to create slanted fractions. Usage: θ

Alternatively, the nicefrac package provides the ± command, whose usage is similar to ±.

分數的代碼格式是: $\frac{分子}{分母}$

繁分數[編輯]

Continued fractions should be written using ± command[3]: θ} \end{equation} |render= }}

方根[編輯]

The ± command creates a square root surrounding an expression. It accepts an optional argument specified in square brackets ([ and ]) to change magnitude: θ \] |render= }} θ


Some people prefer writing the square root "closing" it over its content. This method arguably makes it more clear just what is in the scope of the root sign. This habit is not normally used while writing with the computer, but if you still want to change the output of the square root, LaTeX gives you this possibility. Just add the following code in the preamble of your document: θ

|render=

The new style is on left, the old one on right

}} This TeX code first renames the ± command as ±, then redefines ± in terms of the old one, adding something more. The new square root can be seen in the picture on the right, compared to the old one. Unfortunately this code won't work if you want to use multiple roots: if you try to write as ±} after you used the code above, you'll just get a wrong output. In other words, you can redefine the square root this way only if you are not going to use multiple roots in the whole document.

總和與積分[編輯]

The ± and ± commands insert the sum and integral symbols respectively, with limits specified using the caret (^) and underscore (_): θ θ

There are many other "big" commands which operate in a similar manner:

\sum \prod \coprod
\bigoplus \bigotimes \bigodot
\bigcup \bigcap \biguplus
\bigsqcup \bigvee \bigwedge
\int \oint \iint[3]
\iiint[3] \iiiint[3] \idotsint[3]

For more integral symbols, including those not included by default in the Computer Modern font, try the esint package.

The ± command[3] allows the use of ± to write the limits over multiple lines: θ

P(i,j)

\] |render= }}

If you want the limits of an integral to be specified above and below the symbol (like the sum), use the ± command: θ However if you want this to apply to ALL integrals, it is preferable to specify the \left.\begin{array}{r}

 -\frac{1}{m}=\frac{\sum_{i=1}^n{X_iY_i}-\frac{1}{n}\sum_{i=1}^n{X_i\sum_{i=1}^n{Y_i}}}{\sum_{i=1}^n{X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^n{X_i}\right)^2}}\\
 \frac{1}{m}\lg C=\frac{1}{n}\left(\sum_{i=1}^n{Y_i+\frac{1}{m}\sum_{i=1}^n{X_i}}\right)\\

\end{array} \right\} option when loading the amsmath package: \documentclass{artical} \begin{document} Hallo! \end{document}

Subscripts and superscripts in other contexts as well as other parameters to amsmath package related to them are described in Advanced Mathematics chapter.

For bigger integrals, you may use personal declarations, or the bigints package [4].

Brackets, braces and delimiters[編輯]

How to use braces in multi line equations is described in the Advanced Mathematics chapter.

The use of delimiters such as brackets soon becomes important when dealing with anything but the most trivial equations. Without them, formulas can become ambiguous. Also, special types of mathematical structures, such as matrices, typically rely on delimiters to enclose them.

There are a variety of delimiters available for use in LaTeX: θ

Automatic sizing[編輯]

Very often mathematical features will differ in size, in which case the delimiters surrounding the expression should vary accordingly. This can be done automatically using the ± and ± commands. Any of the previous delimiters may be used in combination with these: θ If a delimiter on only one side of an expression is required, then an invisible delimiter on the other side may be denoted using a period (.).

Manual sizing[編輯]

In certain cases, the sizing produced by the ± and ± commands may not be desirable, or you may simply want finer control over the delimiter sizes. In this case, the ±, ±, ± and ± modifier commands may be used: θ

矩陣與數列[編輯]

A basic matrix may be created using the Template:LaTeX/Environment environment[3]: in common with other table-like structures, entries are specified by row, with columns separated using an ampersand (±) and a new rows separated with a double backslash (±) θ

To specify alignment of columns in the table, use starred version[5]: θ

The alignment by default is \left.\begin{array}{r}

 -\frac{1}{m}=\frac{\sum_{i=1}^n{X_iY_i}-\frac{1}{n}\sum_{i=1}^n{X_i\sum_{i=1}^n{Y_i}}}{\sum_{i=1}^n{X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^n{X_i}\right)^2}}\\
 \frac{1}{m}\lg C=\frac{1}{n}\left(\sum_{i=1}^n{Y_i+\frac{1}{m}\sum_{i=1}^n{X_i}}\right)\\

\end{array} \right\} but it can be any column type valid in Template:LaTeX/Environment environment.

However matrices are usually enclosed in delimiters of some kind, and while it is possible to use the \left and \right commands, there are various other predefined environments which automatically include delimiters:

Environment name Surrounding delimiter Notes
Template:LaTeX/Environment[3] centers columns by default
Template:LaTeX/Environment[5] allows to specify alignment of columns in optional parameter
Template:LaTeX/Environment[3] centers columns by default
Template:LaTeX/Environment[5] allows to specify alignment of columns in optional parameter
Template:LaTeX/Environment[3] centers columns by default
Template:LaTeX/Environment[5] allows to specify alignment of columns in optional parameter
Template:LaTeX/Environment[3] centers columns by default
Template:LaTeX/Environment[5] allows to specify alignment of columns in optional parameter
Template:LaTeX/Environment[3] centers columns by default
Template:LaTeX/Environment[5] allows to specify alignment of colums in optional parameter

When writing down arbitrary sized matrices, it is common to use horizontal, vertical and diagonal triplets of dots (known as ellipses) to fill in certain columns and rows. These can be specified using the ±, ± and ± respectively: θ In some cases you may want to have finer control of the alignment within each column, or want to insert lines between columns or rows. This can be achieved using the Template:LaTeX/Environment environment, which is essentially a math-mode version of the tabular environment, which requires that the columns be pre-specified: θ

You may see that the AMS matrix class of environments doesn't leave enough space when used together with fractions resulting in output similar to this:

To counteract this problem, add additional leading space with the optional parameter to the ± command:

θ

If you need "border" or "indexes" on your matrix, plain TeX provides the macro ± θ


Matrices in running text[編輯]

To insert a small matrix, and not increase leading in the line containing it, use Template:LaTeX/Environment environment:

θ

在方程式中加入文字[編輯]

The math environment differs from the text environment in the representation of text. Here is an example of trying to represent text within the math environment: θ

There are two noticeable problems: there are no spaces between words or numbers, and the letters are italicized and more spaced out than normal. Both issues are simply artifacts of the maths mode, in that it treats it as a mathematical expression: spaces are ignored (LaTeX spaces mathematics according to its own rules), and each character is a separate element (so are not positioned as closely as normal text).

There are a number of ways that text can be added properly. The typical way is to wrap the text with the ±} command [3] (a similar command is ±}, though this causes problems with subscripts, and has a less descriptive name). Let's see what happens when the above equation code is adapted: θ

The text looks better. However, there are no gaps between the numbers and the words. Unfortunately, you are required to explicitly add these. There are many ways to add spaces between maths elements, but for the sake of simplicity you may literally add the space character in the affected ±(s) itself (just before the text.) θ

格式化文字[編輯]

Using the ± is fine and gets the basic result. Yet, there is an alternative that offers a little more flexibility. You may recall the introduction of font formatting commands, such as ±, ±, ±, etc. These commands format the argument accordingly, e.g., ±} gives bold text. These commands are equally valid within a maths environment to include text. The added benefit here is that you can have better control over the font formatting, rather than the standard text achieved with ±.

θ

格式化數學符號[編輯]

So we can format text, what about formatting mathematics? There are a set of formatting commands very similar to the font formatting ones just used, except they are aimed specifically for text in maths mode (requires amsfonts)

LaTeX command Sample Description Common use
±} the default math font most mathematical notation
±} this is the default or normal font, unitalicised units of measurement, one word functions
±} italicised font
±} bold font vectors
±} Sans-serif
±} ABCDEFabcdef123456 Monospace (fixed-width) font
±} Calligraphy (uppercase only) often used for sheaves/schemes and categories
±}[6] Fraktur Almost canonical font for Lie algebras
±}[6] Blackboard bold Used to denote special sets (e.g. real numbers)
±}[7] Script

The maths formatting commands can be wrapped around the entire equation, and not just on the textual elements: they only format letters, numbers, and uppercase Greek, and the rest of the maths syntax is ignored.

To bold lowercase Greek or other symbols use the ± command[3]; this will only work if there exists a bold version of the symbol in the current font. As a last resort there is the ± command[3] (poor mans bold): this prints multiple versions of the character slightly offset against each other θ To change the size of the fonts in math mode, see Changing font size.

重音標記[編輯]

So what to do when you run out of symbols and fonts? Well the next step is to use accents:

a' a'' a''' a''''
\hat{a} \bar{a} \overline{aaa} \check{a} \tilde{a}
\grave{a} \acute{a} \breve{a} \vec{a}
\dot{a} \ddot{a} \dddot{a}[3] \ddddot{a}[3]
\not{a} \mathring{a} \widehat{AAA} \widetilde{AAA}

正負號[編輯]

Latex deals with the + and − signs in two possible ways. The most common is as a binary operator. When two maths elements appear either side of the sign, it is assumed to be a binary operator, and as such, allocates some space either side of the sign. The alternative way is a sign designation. This is when you state whether a mathematical quantity is either positive or negative. This is common for the latter, as in maths, such elements are assumed to be positive unless a − is prefixed to it. In this instance, you want the sign to appear close to the appropriate element to show their association. If you put a + or a − with nothing before it but you want it to be handled like a binary operator you can add an invisible character before the operator using ±}. This can be useful if you are writing multiple-line formulas, and a new line could start with a = or a +, for example, then you can fix some strange alignments adding the invisible character where necessary.

控制水平行距[編輯]

LaTeX is obviously pretty good at typesetting maths—it was one of the chief aims of the core Tex system that LaTeX extends. However, it can't always be relied upon to accurately interpret formulas in the way you did. It has to make certain assumptions when there are ambiguous expressions. The result tends to be slightly incorrect horizontal spacing. In these events, the output is still satisfactory, yet, any perfectionists will no doubt wish to fine-tune their formulas to ensure spacing is correct. These are generally very subtle adjustments.

There are other occasions where LaTeX has done its job correctly, but you just want to add some space, maybe to add a comment of some kind. For example, in the following equation, it is preferable to ensure there is a decent amount of space between the maths and the text.

θ

This code produces errors with Miktex 2.9 and does not yield the results seen on the right. Use \textrm instead of just \text.


(Note that this particular example can be expressed in more elegant code by the Template:LaTeX/Environment construct provided by the amsmath package described in Advanced Mathematics chapter.)

LaTeX has defined two commands that can be used anywhere in documents (not just maths) to insert some horizontal space. They are ± and ±

A ± is a space equal to the current font size. So, if you are using an 11pt font, then the space provided by ± will also be 11pt (horizontally, of course.) The ± gives twice that amount. As you can see from the code from the above example, ±s were used to add some separation between the maths and the text.

OK, so back to the fine tuning as mentioned at the beginning of the document. A good example would be displaying the simple equation for the indefinite integral of y with respect to x:

If you were to try this, you may write:

θ

However, this doesn't give the correct result. LaTeX doesn't respect the white-space left in the code to signify that the y and the dx are independent entities. Instead, it lumps them altogether. A ± would clearly be overkill is this situation—what is needed are some small spaces to be utilized in this type of instance, and that's what LaTeX provides:

Command Description Size
± small space 3/18 of a quad
± medium space 4/18 of a quad
± large space 5/18 of a quad
± negative space -3/18 of a quad

NB you can use more than one command in a sequence to achieve a greater space if necessary.

So, to rectify the current problem:

θ θ θ

The negative space may seem like an odd thing to use, however, it wouldn't be there if it didn't have some use! Take the following example:

θ


The matrix-like expression for representing binomial coefficients is too padded. There is too much space between the brackets and the actual contents within. This can easily be corrected by adding a few negative spaces after the left bracket and before the right bracket.

θ

In any case, adding some spaces manually should be avoided whenever possible: it makes the source code more complex and it's against the basic principles of a What You See is What You Mean approach. The best thing to do is to define some commands using all the spaces you want and then, when you use your command, you don't have to add any other space. Later, if you change your mind about the length of the horizontal space, you can easily change it modifying only the command you defined before. Let us use an example: you want the d of a dx in an integral to be in roman font and a small space away from the rest. If you want to type an integral like ±, you can define a command like this: \documentclass{artical} \begin{document} Hallo! \end{document} }} in the preamble of your document. We have chosen ± just because it reminds the "d" it replaces and it is fast to type. Doing so, the code for your integral becomes ±. Now, whenever you write an integral, you just have to use the ± instead of the "d", and all your integrals will have the same style. If you change your mind, you just have to change the definition in the preamble, and all your integrals will be changed accordingly.

Advanced Mathematics: AMS Math package[編輯]

The AMS (American Mathematical Society) mathematics package is a powerful package that creates a higher layer of abstraction over mathematical LaTeX language; if you use it it will make your life easier. Some commands amsmath introduces will make other plain LaTeX commands obsolete: in order to keep consistency in the final output you'd better use amsmath commands whenever possible. If you do so, you will get an elegant output without worrying about alignment and other details, keeping your source code readable. If you want to use it, you have to add this in the preamble: \documentclass{artical} \begin{document} Hallo! \end{document}

Introducing text and dots in formulas[編輯]

amsmath defines also the ± command, that is a generalization of the existing ±. You can use ± in both text and math mode and LaTeX will replace it with three dots "…" but it will decide according to the context whether to put it on the bottom (like ±) or centered (like ±).

Dots[編輯]

LaTeX gives you several commands to insert dots in your formulas. This can be particularly useful if you have to type big matrices omitting elements. First of all, here are the main dots-related commands LaTeX provides:

Code Output Comment
± generic dots, to be used in text (outside formulas as well). It automatically manages whitespaces before and after itself according to the context, it's a higher level command.
± the output is similar to the previous one, but there is no automatic whitespace management; it works at a lower level.
± These dots are centered relative to the height of a letter. There is also the binary multiplication operator, \cdot, mentioned below.
± vertical dots
± diagonal dots
± inverse diagonal dots (requires the mathdots package)
± to be used in matrices, it creates a row of dots spanning n columns.

Instead of using ± and ±, you should use the semantically oriented commands. It makes it possible to adapt your document to different conventions on the fly, in case (for example) you have to submit it to a publisher who insists on following house tradition in this respect. The default treatment for the various kinds follows American Mathematical Society conventions.

Code Output Comment
± LaTeX Dotsc.png for "dots with commas"
± LaTeX Dotsb.png for "dots with binary operators/relations"
± LaTeX Dotsm.png for "multiplication dots"
± LaTeX Dotsi.png for "dots with integrals"
± LaTeX Dotso.png for "other dots" (none of the above)

數學符號列表[編輯]

All the pre-defined mathematical symbols from the \TeX\ package are listed below. More symbols are available from extra packages.

Relation Symbols
Symbol Script Symbol Script Symbol Script Symbol Script Symbol Script
\leq \geq \equiv \models \prec
\succ \sim \perp \preceq \succeq
\simeq \mid \ll \gg \asymp
\parallel \subset \supset \approx \bowtie
\subseteq \supseteq \cong \sqsubset \sqsupset
\neq \smile \sqsubseteq \sqsupseteq \doteq
\frown \in \ni \propto =
\vdash \dashv < >
Greek Letters
Symbol Script
and \Alpha and \alpha
and \Beta and \beta
and \Gamma and \gamma
and \Delta and \delta
, and \Epsilon, \epsilon and \varepsilon
and \Zeta and \zeta
and \Eta and \eta
, and \Theta, \theta and \vartheta
and \Iota and \iota
and \Kappa and \kappa
and \Lambda and \lambda
and \Mu and \mu
and \Nu and \nu
and \Xi and \xi
, and \Pi, \pi and \varpi
, and \Rho, \rho and \varrho
, and \Sigma, \sigma and \varsigma
and \Tau and \tau
and \Upsilon and \upsilon
, , and \Phi, \phi and \varphi
and \Chi and \chi
and \Psi and \psi
and \Omega and \omega
Binary Operations
Symbol Script Symbol Script Symbol Script Symbol Script
\pm \cap \diamond \oplus
\mp \cup \bigtriangleup \ominus
\times \uplus \bigtriangledown \otimes
\div \sqcap \triangleleft \oslash
\ast \sqcup \triangleright \odot
\star \vee \bigcirc \circ
\wedge \dagger \bullet \setminus
\ddagger \cdot \wr \amalg
Set and/or Logic Notation
Symbol Script
\exists
\forall
\neg
and \in and \notin
\ni
\land
\lor
\rightarrow
\implies
\iff
\top
\bot
and \emptyset and \varnothing
Delimiters
Symbol Script
\uparrow
\Uparrow
\downarrow
\Downarrow
\{
\}
\lceil
\rceil
\langle
\rangle
/
\backslash
|
\|
Other symbols
Symbol Script
\partial
\infty
\nabla
\hbar
\Box
\aleph
\ell
\imath
\jmath
\Re
\Im


Trigonometric Functions
Symbol Script Symbol Script Symbol Script Symbol Script
\sin \cos \tan \cot
\arcsin \arccos \arctan \arccot
\sinh \conh \tanh \coth
\sec \csc


總結[編輯]

As you begin to see, typesetting math can be tricky at times. However, because Latex provides so much control, you can get professional quality mathematics typesetting with relatively little effort (once you've had a bit of practice, of course!). It would be possible to keep going and going with math topics because it seems potentially limitless. However, with this tutorial, you should be able to get along sufficiently.

TODO

編者待辦:

  • introduce symbols from [1]
  • add symbols from [2]
  • consider adding symbols from [3] -- the list of nearly all symbols available for LaTeX
  • Consider, instead of using the symbols from the above mentioned, using what has already been introduced in [4] instead of retyping the tables
  • How to box an equation within an align environment
  • Color in equations

附記[編輯]

  1. http://www.ams.org/publications/authors/tex/amslatex
  2. http://www.tex.ac.uk/ctan/macros/latex/contrib/mh/mathtools.pdf
  3. 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 requires the amsmath package
  4. http://hdl.handle.net/2268/6219
  5. 5.0 5.1 5.2 5.3 5.4 5.5 requires the mathtools package
  6. 6.0 6.1 requires amsfonts or amssymb packages
  7. require mathrsfs package

延伸閱讀[編輯]

其他連結[編輯]


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