LaTeX数学符号大全.pdf

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Short Math Guide for LATEX Michael Downes American Mathematical Society Version 1.09 (2002-03-22), currently available at http://www.ams.org/tex/short-math-guide.html 1. Introduction This is a concise summary of recommended features in LATEX and a couple of extension packages for writing math formulas. Readers needing greater depth of detail are referred to the sources listed in the bibliography, especially [Lamport], [LUG], [AMUG], [LFG], [LGG], and [LC]. A certain amount of familiarity with standard LATEX terminology is assumed; if your memory needs refreshing on the LATEX meaning of command, optional argument, environment, package, and so forth, see [Lamport]. The features described here are available to you if you use LATEX with two extension packages published by the American Mathematical Society: amssymb and amsmath. Thus, the source ﬁle for this document begins with \documentclass{article} \usepackage{amssymb,amsmath} The amssymb package might be omissible for documents whose math symbol usage is rela- tively modest; the easiest way to test this is to leave out the amssymb reference and see if any math symbols in the document produce ‘Undeﬁned control sequence’ messages. Many noteworthy features found in other packages are not covered here; see Section 10. Regarding math symbols, please note especially that the list given here is not intended to be comprehensive, but to illustrate such symbols as users will normally ﬁnd already present in their LATEX system and usable without installing any additional fonts or doing other setup work. If you have a need for a symbol not shown here, you will probably want to consult The Comprehensive LATEX Symbols List (Pakin): http://www.ctan.org/tex-archive/info/symbols/comprehensive/. 2. Inline math formulas and displayed equations 2.1. The fundamentals Entering and leaving math mode in LATEX is normally done with the following commands and environments. inline formulas displayed equations $ . . . $ $ . . . $ \[...\] \begin{equation*} . . . \end{equation*} \begin{equation} . . . \end{equation} unnumbered unnumbered automatically numbered Note. Alternative environments \begin{math} . . . \end{math}, \begin{displaymath} . . . \end{displaymath} are seldom needed in practice. Using the plain TEX notation $$ . . . $$ for displayed equations is not recom- mended. Although it is not expressly forbidden in LATEX, it is not documented anywhere in the LATEX book as being part of the LATEX command set, and it interferes with the proper operation of various features such as the fleqn option. Environments for handling equation groups and multi-line equations are shown in Table 1. 1

Short Math Guide for LATEX, version 1.09 (2002-03-22) 2 Table 1: Multi-line equations and equation groups (vertical lines indicating nominal mar- gins). \begin{equation}\label{xx} \begin{split} a& =b+c-d\\ & \quad +e-f\\ & =g+h\\ & =i \end{split} \end{equation} \begin{multline} a+b+c+d+e+f\\ +i+j+k+l+m+n \end{multline} \begin{gather} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather} \begin{align} a_1& =b_1+c_1\\ a_2& =b_2+c_2-d_2+e_2 \end{align} \begin{align} a_{11}& =b_{11}& a_{12}& =b_{12}\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \end{align} \begin{flalign*} a_{11}& =b_{11}& a_{12}& =b_{12}\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \end{flalign*} a = b + c − d + e − f = g + h = i (2.1) a + b + c + d + e + f + i + j + k + l + m + n (2.2) a1 = b1 + c1 a2 = b2 + c2 − d2 + e2 a1 = b1 + c1 a2 = b2 + c2 − d2 + e2 (2.3) (2.4) (2.5) (2.6) a11 = b11 a21 = b21 a12 = b12 a22 = b22 + c22 (2.7) (2.8) a11 = b11 a21 = b21 a12 = b12 a22 = b22 + c22 Note 1. The split environment is something of a special case. It is a subordinate environment that can be used as the contents of an equation environment or the contents of one “line” in a multiple-equation structure such as align or gather. Note 2. The eqnarray and eqnarray* environments described in [Lamport] are not recommended because they produce inconsistent spacing of the equal signs and make no attempt to prevent overprinting of the equation body and equation number.

Short Math Guide for LATEX, version 1.09 (2002-03-22) 2.2. Automatic numbering and cross-referencing To get an auto-numbered equa- tion, use the equation environment; to assign a label for cross-referencing, use the \label command: 3 \begin{equation}\label{reio} ... \end{equation} To get a cross-reference to an auto-numbered equation, use the \eqref command: ... using equations \eqref{ax1} and \eqref{bz2}, we can derive ... The above example would produce something like using equations (3.2) and (3.5), we can derive In other words, \eqref{ax1} is equivalent to (\ref{ax1}). To give your equation numbers the form m.n (section-number.equation-number), use the \numberwithin command in the preamble of your document: \numberwithin{equation}{section} For more details on custom numbering schemes see [Lamport, §6.3, §C.8.4]. The subequations environment provides a convenient way to number equations in a group with a subordinate numbering scheme. For example, supposing that the current equation number is 2.1, write \begin{equation}\label{first} a=b+c \end{equation} some intervening text \begin{subequations}\label{grp} \begin{align} a&=b+c\label{second}\\ d&=e+f+g\label{third}\\ h&=i+j\label{fourth} \end{align} \end{subequations} to get some intervening text a = b + c a = b + c d = e + f + g h = i + j (2.9) (2.10a) (2.10b) (2.10c) By putting a \label command immediately after \begin{subequations} you can get a reference to the parent number; \eqref{grp} from the above example would produce (2.10) while \eqref{second} would produce (2.10a). 3. Math symbols and math fonts 3.1. Classes of math symbols The symbols in a math formula fall into diﬀerent classes that correspond more or less to the part of speech each symbol would have if the formula were expressed in words. Certain spacing and positioning cues are traditionally used for the diﬀerent symbol classes to increase the readability of formulas.

Short Math Guide for LATEX, version 1.09 (2002-03-22) 4 Class number Mnemonic 0 1 2 3 4 5 6 Ord Op Bin Rel Open Close Pun Examples A 0 Φ ∞ P Q R Description (part of speech) simple/ordinary (“noun”) preﬁx operator binary operator (conjunction) + ∪ ∧ = < ⊂ relation/comparison (verb) ( [ { h left/opening delimiter ) ] } i right/closing delimiter postﬁx/punctuation . , ; ! Note 1. The distinction in TEX between class 0 and an additional class 7 has to do only with font selection issues and is immaterial here. Note 2. Symbols of class Bin, notably the minus sign −, are automatically coerced to class 0 (no space) if they do not have a suitable left operand. The spacing for a few symbols follows tradition instead of the general rule: although / is (semantically speaking) of class 2, we write k/2 with no space around the slash rather than k / 2. And compare p|q p|q (no space) with p\mid q p | q (class-3 spacing). The proper way to deﬁne a new math symbol is discussed in LATEX 2ε font selection [LFG]. It is not really possible to give a useful synopsis here because one needs ﬁrst to understand the ramiﬁcations of font speciﬁcations. 3.2. Some symbols intentionally omitted here The following math symbols that are mentioned in the LATEX book [Lamport] are intentionally omitted from this discussion because they are superseded by equivalent symbols when the amssymb package is loaded. If you are using the amssymb package anyway, the only thing that you are likely to gain by using the alternate name is an unnecessary increase in the number of fonts used by your document. \Box, see \square \Diamond, see \lozenge ♦ \leadsto, see \rightsquigarrow \Join, see \bowtie ./ \lhd, see \vartriangleleft C \unlhd, see \trianglelefteq E \rhd, see \vartriangleright B \unrhd, see \trianglerighteq D Furthermore, there are many, many additional symbols available for LATEX use above and beyond the ones included here. This list is not intended to be comprehensive. For a much more comprehensive list of symbols, including nonmathematically oriented ones such as phonetic alphabetic or dingbats, see The Comprehensive LATEX Symbols List (Pakin): http://www.ctan.org/tex-archive/info/symbols/comprehensive/. 3.3. Latin letters and Arabic numerals The Latin letters are simple symbols, class 0. The default font for them in math formulas is italic. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z When adding an accent to an i or j in math, dotless variants can be obtained with \imath and \jmath: ı \imath  \jmath ˆ \hat{\jmath} Arabic numerals 0–9 are also of class 0. Their default font is upright/roman. 0 1 2 3 4 5 6 7 8 9

Short Math Guide for LATEX, version 1.09 (2002-03-22) 3.4. Greek letters Like the Latin letters, the Greek letters are simple symbols, class 0. For obscure historical reasons, the default font for lowercase Greek letters in math formu- las is italic while the default font for capital Greek letters is upright/roman. (In other ﬁelds such as physics and chemistry, however, the typographical traditions are somewhat diﬀerent.) The capital Greek letters not present in this list are the letters that have the same appearance as some Latin letter: A for Alpha, B for Beta, and so on. In the list of lowercase letters there is no omicron because it would be identical in appearance to Latin o. In practice, the Greek letters that have Latin look-alikes are seldom used in math formulas, to avoid confusion. 5 Γ \Gamma ∆ \Delta Λ \Lambda Φ \Phi Π \Pi Ψ \Psi Σ \Sigma Θ \Theta Υ \Upsilon Ξ \Xi Ω \Omega α \alpha β \beta γ \gamma δ \delta \epsilon ζ \zeta η \eta θ \theta ι \iota κ \kappa λ \lambda µ \mu ν \nu ξ \xi π \pi ρ \rho σ \sigma τ \tau υ \upsilon φ \phi χ \chi ψ \psi ω \omega z \digamma ε \varepsilon κ \varkappa ϕ \varphi \varpi \varrho ς \varsigma ϑ \vartheta 3.5. Other alphabetic symbols These are also class 0. ℵ \aleph \hslash f \mho i \beth k \daleth ∂ \partial ג \gimel ℘ \wp { \complement ‘ \ell ð \eth \hbar s \circledS k \Bbbk ‘ \Finv a \Game = \Im< \Re 3.6. Miscellaneous simple symbols These symbols are also of class 0 (ordinary) which means they do not have any built-in spacing. ♣ \clubsuit \diagdown \diagup ♦ \diamondsuit ∅ \emptyset ∃ \exists [ \flat ∀ \forall ♥ \heartsuit ∞ \infty # \# & \& ∠ \angle 8 \backprime F \bigstar \blacklozenge \blacksquare N \blacktriangle H \blacktriangledown ⊥ \bot Note 1. A common mistake in the use of the symbols and # is to try to make them serve as binary operators or relation symbols without using a properly deﬁned math symbol command. If you merely use the existing commands \square or \# the inter-symbol spacing will be incorrect because those commands produce a class-0 symbol. Note 2. Synonyms: ¬ \lnot \square √ \surd > \top 4 \triangle O \triangledown ∅ \varnothing ♦ \lozenge ] \measuredangle ∇ \nabla \ \natural ¬ \neg @ \nexists 0 \prime ] \sharp ♠ \spadesuit ^ \sphericalangle

Short Math Guide for LATEX, version 1.09 (2002-03-22) 3.7. Binary operator symbols ∗ * + +− -q \amalg · \cdot \centerdot ◦ \circ ~ \circledast } \circledcirc \circleddash ∪ \cup d \Cup g \curlyvee f \curlywedge † \dagger ‡ \ddagger \diamond ÷ \div > \divideontimes u \dotplus [ \doublebarwedge m \gtrdot | \intercal h \leftthreetimes l \lessdot n \ltimes ∓ \mp \odot \ominus ⊕ \oplus \oslash ⊗ \otimes ± \pm i \rightthreetimes o \rtimes \ \setminus ∗ \ast Z \barwedge \bigcirc 5 \bigtriangledown 4 \bigtriangleup \boxdot \boxminus \boxplus \boxtimes • \bullet ∩ \cap e \Cap Synonyms: ∧ \land, ∨ \lor, d \doublecup, e \doublecap 3.8. Relation symbols: < = > ∼ and variants 5 \leqq 6 \leqslant / \lessapprox Q \lesseqgtr S \lesseqqgtr ≶ \lessgtr . \lesssim \ll ≪ \lll \lnapprox \lneq \lneqq \lnsim \lvertneqq \ncong 6= \neq \ngeq \ngeqq \ngeqslant 0 \eqslantless ≡ \equiv ; \fallingdotseq ≥ \geq = \geqq > \geqslant \gg ≫ \ggg \gnapprox \gneq \gneqq \gnsim ’ \gtrapprox R \gtreqless T \gtreqqless ≷ \gtrless & \gtrsim \gvertneqq ≤ \leq < < = = > > ≈ \approx u \approxeq \asymp v \backsim w \backsimeq l \bumpeq m \Bumpeq $ \circeq ∼= \cong 2 \curlyeqprec 3 \curlyeqsucc .= \doteq + \doteqdot P \eqcirc h \eqsim 1 \eqslantgtr Synonyms: 6= \ne, ≤ \le, ≥ \ge, + \Doteq, ≪ \llless, ≫ \gggtr 6 r \smallsetminus u \sqcap t \sqcup ? \star × \times / \triangleleft . \triangleright ] \uplus ∨ \vee Y \veebar ∧ \wedge o \wr ∼ \sim ’ \simeq \succ v \succapprox < \succcurlyeq \succeq \succnapprox \succneqq \succnsim % \succsim ≈ \thickapprox ∼ \thicksim , \triangleq ≯ \ngtr \nleq \nleqq \nleqslant ≮ \nless ⊀ \nprec \npreceq \nsim \nsucc \nsucceq ≺ \prec w \precapprox 4 \preccurlyeq \preceq \precnapprox \precneqq \precnsim - \precsim : \risingdotseq

Short Math Guide for LATEX, version 1.09 (2002-03-22) 3.9. Relation symbols: arrows See also Section 4. \circlearrowleft W \Lleftarrow ←− \longleftarrow \circlearrowright ⇐= \Longleftarrow x \curvearrowleft ←→ \longleftrightarrow y \curvearrowright ⇐⇒ \Longleftrightarrow \downdownarrows 7−→ \longmapsto \downharpoonleft −→ \longrightarrow \downharpoonright ←- \hookleftarrow =⇒ \Longrightarrow ,→ \hookrightarrow " \looparrowleft ← \leftarrow # \looparrowright ⇐ \Leftarrow \Lsh 7→ \mapsto \leftarrowtail ( \multimap ) \leftharpoondown : \nLeftarrow ( \leftharpoonup ⇔ \leftleftarrows < \nLeftrightarrow ↔ \leftrightarrow ; \nRightarrow ⇔ \Leftrightarrow % \nearrow \leftrightarrows 8 \nleftarrow \leftrightharpoons = \nleftrightarrow ! \leftrightsquigarrow 9 \nrightarrow Synonyms: ← \gets, → \to, \restriction 3.10. Relation symbols: miscellaneous 7 - \nwarrow → \rightarrow ⇒ \Rightarrow \rightarrowtail + \rightharpoondown * \rightharpoonup \rightleftarrows \rightleftharpoons ⇒ \rightrightarrows \rightsquigarrow V \Rrightarrow \Rsh& \searrow . \swarrow \twoheadleftarrow \twoheadrightarrow \upharpoonleft \upharpoonright \upuparrows \backepsilon ∵ \because G \between J \blacktriangleleft I \blacktriangleright ./ \bowtie a \dashv _ \frown ∈ \in | \mid |= \models 3 \ni - \nmid /∈ \notin ∦ \nparallel . \nshortmid / \nshortparallel * \nsubseteq Synonyms: 3 \owns " \nsubseteqq + \nsupseteq # \nsupseteqq 6 \ntriangleleft 5 \ntrianglelefteq 7 \ntriangleright 4 \ntrianglerighteq 0 \nvdash 1 \nVdash 2 \nvDash 3 \nVDash k \parallel ⊥ \perp t \pitchfork ∝ \propto p \shortmid q \shortparallel a \smallfrown ‘ \smallsmile ^ \smile @ \sqsubset v \sqsubseteq A \sqsupset w \sqsupseteq ⊂ \subset b \Subset ⊆ \subseteq j \subseteqq ( \subsetneq $ \subsetneqq ⊃ \supset c \Supset ⊇ \supseteq k \supseteqq ) \supsetneq % \supsetneqq ∴ \therefore E \trianglelefteq D \trianglerighteq ∝ \varpropto \varsubsetneq & \varsubsetneqq ! \varsupsetneq ’ \varsupsetneqq M \vartriangle C \vartriangleleft B \vartriangleright ‘ \vdash \Vdash \vDash \Vvdash 3.11. Cumulative (variable-size) operators R \int H \oint T \bigcap S \bigcup J \bigodot L \bigoplus N \bigotimes F \bigsqcup U \biguplus W \bigvee V \bigwedge ‘ \coprod Q \prod P \sum ∫ \smallint

Short Math Guide for LATEX, version 1.09 (2002-03-22) 3.12. Punctuation . . / /| | , , ; ; : \colon : : ! ! ? ? ··· \dotsb . . . \dotsc ··· \dotsi ··· \dotsm . . . \dotso ... \ddots 8 ... \vdots Note 1. The : by itself produces a colon with class-3 (relation) spacing. The command \colon produces special spacing for use in constructions such as f\colon A\to B f : A → B. Note 2. Although the commands \cdots and \ldots are frequently used, we recommend the more seman- tically oriented commands \dotsb \dotsc \dotsi \dotsm \dotso for most purposes (see 4.6). 3.13. Pairing delimiters (extensible) See Section 6 for more information.  \lgroup \rgroup  \lmoustache \rmoustache [ ] ( ) hi no \lvert \rvert \Vert \vert \lbrace \rbrace \langle \rangle \lVert \rVert DE lm jk / \lceil \rceil \backslash \lfloor \rfloor . / 3.14. Nonpairing extensible symbols  \arrowvert www \Arrowvert  \bracevert Note 1. Using \vert, |, \Vert, or \| for paired delimiters is not recommended (see 6.2). Synonyms: k \| 3.15. Extensible vertical arrows x \uparrow ~ww \Uparrow y \downarrow ww \Downarrow xy \updownarrow ~w \Updownarrow 3.16. Accents ´x \acute{x} `x \grave{x} ¨x \ddot{x} ˜x \tilde{x} ¯x \bar{x} ˘x \breve{x} ˇx \check{x} ˆx \hat{x} ~x \vec{x} ˙x \dot{x} ¨x \ddot{x} ... x \dddot{x} gxxx \widetilde{xxx} dxxx \widehat{xxx} 3.17. Named operators These operators are represented by a multiletter abbreviation. arccos \arccos arcsin \arcsin arctan \arctan arg \arg cos \cos cosh \cosh cot \cot coth \coth csc \csc deg \deg det \det dim \dim exp \exp gcd \gcd hom \hom inf \inf inj lim \injlim ker \ker lg \lg lim \lim lim inf \liminf lim sup \limsup ln \ln log \log proj lim \projlim max \max min \min Pr \Pr sec \sec sin \sin sinh \sinh sup \sup tan \tan tanh \tanh lim−→ \varinjlim lim←− \varprojlim lim \varliminf lim \varlimsup To deﬁne additional named operators outside the above list, use the \DeclareMathOperator command; for example, after

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