﻿ TeX Commands available in MathJax

# $\rm\TeX$ Commands available in MathJax

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This document was created in Spring of 2011.
As of May 2017, it is being processed using MathJax 2.7.1 (loaded from my own server).

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I (Dr. Carol JVF Burns) prepared this page to thoroughly familiarize myself with the $\rm\TeX$ commands that are available in MathJax,
and to provide a resource that may be useful to other MathJax users.
Davide Cervone, the lead developer of MathJax, has most generously provided extensive edits,
and this page is greatly improved due to his efforts; I owe him countless thanks.
All mistakes on this page are my own (and I welcome suggestions and corrections).

MathJax allows a syntax modeled on both $\rm\TeX$ and $\rm\LaTeX$.
Therefore, web authors can use familiar and concise commands when creating mathematics with MathJax.

# Alphabetical List of $\rm\TeX$ Commands available in MathJax

 symbols 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 environments

Know the shape of a character that you want, but not its name?   Draw it here!

symbols
#   indicates numbered arguments in definitions

Example:
 \def\specialFrac#1#2{\frac{x + #1}{y + #2}} \specialFrac{7}{z+3}  yields $$\def\specialFrac#1#2{\frac{x + #1}{y + #2}} \specialFrac{7}{z+3}$$
%   used for a single-line comment;
shows only in the source code;
does not show in the rendered expression

Example (showing the math block delimiters):
 $$% Note: (x+1)^2 is NOT x^2 + 1 (x+1)^2 % original expression = (x+1)(x+1) % definition of exponent = x^2 + 2x + 1 % FOIL, combine like terms$$  yields $$% Note: (x+1)^2 is NOT x^2 + 1 (x+1)^2 % original expression = (x+1)(x+1) % definition of exponent = x^2 + 2x + 1 % FOIL, combine like terms$$
Internet Explorer caution:
&   used as separators in alignment environments;
used in HTML entity references within math mode;
for a literal ampersand, use \&

Examples:
 \begin{matrix} a & b\cr c & d \end{matrix} yields $\begin{matrix} a & b\cr c & d \end{matrix}$ a < b yields $a ^ used to indicate exponents; used to indicate superscripts; used for limits on large operators and in some ‘vertical’ constructions (see examples) <optional #1> ^ #2 argument #1 is optional; use braces, as needed, to clarify what is the exponent Examples:  ^i yields$^i$x^i_2 yields$x^i_2${x^i}_2 yields${x^i}_2$x^{i_2} yields$x^{i_2}$x^{i^2} yields$x^{i^2}${x^i}^2 yields${x^i}^2$Note: x^i^2 yields an error. ^ax^b yields$^ax^b$\sum_{n=1}^\infty yields$\sum_{n=1}^\infty$(inline mode) \overbrace{x+\cdots+x} ^{n\text{ times}} yields$\overbrace{x+\cdots+x} ^{n\text{ times}}$_ used to indicate subscripts; used for limits on large operators and in some ‘vertical’ constructions (see examples) <optional #1> _ #2 argument #1 is optional; use braces, as needed, to clarify what is the subscript Examples:  _2 yields$_2$x_i^2 yields$x_i^2${x_i}^2 yields${x_i}^2$x_{i^2} yields$x_{i^2}$x_{i_2} yields$x_{i_2}${x_i}_2 yields${x_i}_2$Note: x_i_2 yields an error. ^a_bx^c_d yields$^a_bx^c_d$\sum_{n=1}^\infty yields$\sum_{n=1}^\infty$(inline mode) \underbrace{x+\cdots+x} _{n\text{ times}} yields$\underbrace{x+\cdots+x} _{n\text{ times}}${ } braces, used for grouping; for literal braces, use \{ and \} There are two basic grouping constructs that use braces; I will refer to them as ‘arguments’ versus ‘braced groups’. If you're not aware which construct is in force, then you can get unexpected results. The examples below should clarify. ARGUMENTS: In this documentation, arguments are indicated by #1, #2, etc. An argument is either a single ‘token’ (like ‘a’ or ‘\alpha’), or is a group enclosed in braces. For example, the \boldsymbol command takes an argument, notated by: \boldsymbol #1 Thus:  \boldsymbol aa yields$\boldsymbol aa$the first token, ‘a’, becomes bold \boldsymbol \alpha\alpha yields$\boldsymbol \alpha\alpha$the first token, ‘\alpha’, becomes bold \boldsymbol{a\alpha}a\alpha yields$\boldsymbol{a\alpha}a\alpha$braces have been used to make the argument the group ‘a\alpha’, so both become bold BRACED GROUPS: A ‘braced group’ is a group, enclosed by braces, inside which some behavior is in force. The \bf (boldface) command operates inside a braced group, notated by: {\bf ... } Here, \bf is a switch, which ‘turns on’ boldface inside the braced group; boldface ends when the braced group ends. Sometimes, you may not see the opening ‘{’ that signals the start of a braced group. In this situation, when does a command (like \bf ) end? It ends at whichever occurs first: • it is replaced by a competing command (e.g., \bf is replaced by \rm ) • the end of math mode (math delimiters form an implicit local group) Examples: (explicit braced groups are indicated in red, for your convenience)  \bf ab yields$\bf ab$turn on boldface;stays on to end of math mode {\bf ab}cd yields${\bf ab}cd$an explicit braced group is entered; the ‘cd’ falls outside this group \bf{ab}cd yields$\bf{ab}cd$turn on boldface;stays on to end of math mode;the braces here are extraneous {\bf{ab}c}d yields${\bf{ab}c}d$boldface operates inside a braced group; the ‘d’ falls outside this group {efg\bf{ab}c}d yields${efg\bf{ab}c}d$the ‘efg’ occur before boldface is turned on ab \bf cd \rm ef yields$ab \bf cd \rm ef$the competing \rm replaces boldface ab \bf cd {\rm ef} gh yields$ab \bf cd {\rm ef} gh$the ‘gh’ is still in boldface Make sure you see the difference in the behaviors below:  \boldsymbol{ab}cd yields$\boldsymbol{ab}cd$\boldsymbol takes an argument \bf{ab}cd yields$\bf{ab}cd$\bf does not take an argument; instead, \bf ‘turns on’ boldface behavior \! negative thin space; i.e., it ‘back ups’ a thin space amount Examples:  \rm IR yields$\rm IR$\rm I\! R yields$\rm I\! R$see also: \negthinspace  \, \: \> \;  \, thin space (normally$\frac 16 = \frac{3}{18}$of a quad) \: medium space (normally$\frac 29 = \frac{4}{18}$of a quad) \> alternate medium space \; thick space (normally$\frac 5{18}$of a quad) Examples:  normal spacing between letters:$abababab$using \, between letters:$a\,b\,a\,b\,a\,b\,a\,b$using \: between letters:$a\:b\:a\:b\:a\:b\:a\:b$using \> between letters:$a\>b\>a\>b\>a\>b\>a\>b$using \; between letters:$a\;b\;a\;b\;a\;b\;a\;b$see also: \thinspace \ (backslash space)  control space;$\rm\TeX$often ignores spaces, or collapses multiple spaces to a single space. A control space is used to force$\rm\TeX$to typeset a space. class ORD Examples:  \rm This is a sentence. yields$\rm This is a sentence.$\rm This\ is\ a\ sentence. yields$\rm This\ is\ a\ sentence.$\rm This~is~a~sentence. yields$\rm This~is~a~sentence.$\text{This is a sentence.} yields$\text{This is a sentence.}$in MathJax, this is the same as: \nobreakspace, \space, ~ (tilde character) see also: \text ~ (tilde character)  In$\rm\TeX$this is a non-breaking space—i.e., a blank space where$\rm\TeX$is not allowed to break between lines. MathJax (unlike$\rm\TeX$) doesn't do any automatic breaking of lines, so MathJax will not break at any space. The tilde is useful to force a space where MathJax would otherwise collapse or ignore spaces, as illustrated in the examples below. class ORD Click here to see examples of what happens with very long math in MathJax. Examples:  \rm Dr. Carol J.V. Fisher yields$\rm Dr. Carol J.V. Fisher$\rm Dr.~Carol~J.V.~Fisher yields$\rm Dr.~Carol~J.V.~Fisher$\text{Dr. Carol J.V. Fisher} yields$\text{Dr. Carol J.V. Fisher}$a b c d yields$a b c d$a~b~~~~~~c~d yields$a~b~~~~~~c~d$in MathJax, this is the same as: \nobreakspace, \space, \  (backslash space) \#$\#$ literal number sign; literal pound sign; needed since # is used to indicate arguments in definitions # class ORD \\$ $\$ $ literal dollar sign; needed since $   may (optionally) be used to delimit math mode Dollar sign outside of math mode: $ class ORD \%$\%$ literal percent sign; needed since % is used to begin a single-line comment % class ORD \&$\& literal ampersand; needed since ampersands are used as separators in alignment environments and for HTML entity references inside math mode & class ORD see also: \And \\ line separator in alignment modes and environments Example:  \begin{gather}a\\a+b\\a+b+c\end{gather} yields\begin{gather}a\\a+b\\a+b+c\end{gather}$For a literal backslash, see \backslash. in MathJax, these are essentially the same: \cr, \newline \_$\_$ literal underscore; needed since underscores are used for subscripts _ class ORD Examples:  a_2 yields$a_2$a\_2 yields$a\_2$\{ \}$\{\ \}$ literal braces; needed since braces are used for grouping in math mode; non-stretchy when used alone; stretchy when used with \left or \right \{ is class OPEN \} is class CLOSE Examples:  {1,2,3} yields${1,2,3}$\{1,2,3\} yields$\{1,2,3\}$\left\{\frac ab,c\right\} yields$\left\{\frac ab,c\right\}$see also: \brace, \lbrace, \rbrace |$|$ pipe character; vertical bar; absolute value; non-stretchy when used alone; stretchy when used with \left or \right class ORD Examples:  |x| yields$|x|$|\frac ab| yields$|\frac ab|$\left|\frac ab\right| yields$\left|\frac ab\right|$\{x | x\in\Bbb Z\} yields$\{x | x\in\Bbb Z\}$\{x\,|\,x\in\Bbb Z\} yields$\{x\,|\,x\in\Bbb Z\}$see also: \lvert, \rvert, \vert \|$\|$ double pipe character; double vertical bar; norm; non-stretchy when used alone; stretchy when used with \left or \right ∥ class ORD Examples:  \|x\| yields$\|x\|$\|\frac ab\| yields$\|\frac ab\|$\left\|\frac ab\right\| yields$\left\|\frac ab\right\|$see also: \lVert, \rVert, \Vert ( )$(\ )$ parentheses; non-stretchy when used alone; stretchy when used with \left or \right ( is class OPEN; ) is class CLOSE Examples:  (\frac ab,c) yields$(\frac ab,c)$\left(\frac ab,c\right) yields$\left(\frac ab,c\right)$.$.$ period; decimal point class PUNCT In some math environments (but not all):  With numbers on either side, there is no surrounding space: 3.14 yields$3.14$With non-numeric characters, there is a slight amount of space on right: a.b yields$a.b$To suppress this space, enclose the ‘.’ in braces: a{.}b yields$a{.}b$/$/$ forward slash; can be used to denote division class ORD Example:  a/b yields$a/b$+$+$ plus symbol; e.g., used for addition class BIN Example:  a+b yields$a+b$-$-$ minus symbol; e.g., used for subtraction class BIN Example:  a-b yields$a-b$-b yields$-b$in most cases, proper spacing is achievedto denote an opposite \text{first: } -a\star b yields$\text{first: } -a\star b$an unusual situation;spacing is not optimal \text{first: } {-}a\star b yields$\text{first: } {-}a\star b$in such cases, you can put the minus sign(or, the group -a ) inside bracesto suppress extra space [ ]$[\ ]$ (square) brackets; non-stretchy when used alone; stretchy when used with \left or \right [ is class OPEN; ] is class CLOSE Examples:  [\frac ab,c] yields$[\frac ab,c]$\left[\frac ab,c\right] yields$\left[\frac ab,c\right]$see also: \brack, \lbrack, \rbrack =$=$ equal; equals class REL see also: \ne, \neq '$'$ prime symbol class ORD Example:  f(x) = x^2,\ f'(x) = 2x,\ f''(x) = 2 yields$f(x) = x^2,\ f'(x) = 2x,\ f''(x) = 2$see also: \prime A \above general command for making fractions; gives control over thickness of horizontal fraction bar { <subformula1> \above <dimen> <subformula2> } Creates a fraction: numerator: subformula1 denominator: subformula2 fraction bar has thickness: dimen There are separate local groups for subformula1 and subformula2 ; if these local groups are not explicit, then unexpected results may occur, as illustrated in the choose discussion. Examples:  a+1 \above 1pt b yields$a+1 \above 1pt b$a \above 1pt b+2 yields$a \above 1pt b+2${a+1 \above 1.5pt b+2}+c yields${a+1 \above 1.5pt b+2}+c$see also: \abovewithdelims, \atop, \atopwithdelims, \cfrac, \dfrac, \frac, \genfrac, \over, \overwithdelims \abovewithdelims general command for making fractions; gives control over thickness of horizontal fraction bar; specifies left and right enclosing delimiters { <subformula1> \abovewithdelims <delim1> <delim2> <dimen> <subformula2> } Creates a fraction: numerator: subformula1 denominator: subformula2 fraction bar has thickness: dimen delim1 is put before the fraction delim2 is put after the fraction For an empty delimiter, use ‘.’ in place of the delimiter. There are separate local groups for subformula1 and subformula2 ; if these local groups are not explicit, then unexpected results may occur, as illustrated in the choose discussion. Examples:  a+1 \abovewithdelims [ ] 1pt b yields$a+1 \abovewithdelims [ ] 1pt b${a \abovewithdelims . | 1.5pt b+2}_{a=3} yields${a \abovewithdelims . | 1.5pt b+2}_{a=3}${a+1 \abovewithdelims \{ \} 1pt b+2}+c yields${a+1 \abovewithdelims \{ \} 1pt b+2}+c$see also: \above, \atop, \atopwithdelims, \cfrac, \dfrac, \frac, \genfrac, \over, \overwithdelims \acute$\acute{}$&#x02CA; acute accent \acute #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \acute e yields$\acute e$\acute E yields$\acute E$\acute eu yields$\acute eu$\acute{eu} yields$\acute{eu}$\aleph$\aleph$ Hebrew letter aleph; commonly used for the cardinality of the real numbers ℵ class ORD \alpha$\alpha$ lowercase Greek letter alpha α class ORD \amalg$\amalg$ this symbol is often used for co-products ⨿ class BIN \And$\And$ ampersand & class ORD see also: \& \angle$\angle$ ∠ class ORD \approx$\approx$ ≈ class REL  \approxeq AMSsymbols$\approxeq$ ≊ class REL \arccos$\arccos$ does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples If alternate notation is desired, define: \def\arccosAlt{\cos^{-1}} so that $\arccosAlt(x)$ yields$\def\arccosAlt{\cos^{-1}} \arccosAlt(x)$class OP \arcsin$\arcsin$ does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples If alternate notation is desired, define: \def\arcsinAlt{\sin^{-1}} so that $\arcsinAlt(x)$ yields$\def\arcsinAlt{\sin^{-1}} \arcsinAlt(x)$class OP \arctan$\arctan$ does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples If alternate notation is desired, define: \def\arctanAlt{\tan^{-1}} so that $\arctanAlt(x)$ yields$\def\arctanAlt{\tan^{-1}} \arctanAlt(x)$class OP \arg$\arg the complex argument function; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples class OP \array a synonym for \matrix \array{ [itex] & [itex] ... \cr <repeat as needed> } alignment occurs at the ampersands; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional Example:  \array{ a & b+1 \cr c+1 & d } yields\array{ a & b+1 \cr c+1 & d }$see also: \matrix \arrowvert$\arrowvert$ not intended for direct use; used internally to create stretchy delimiters ⏐ class ORD see also: |, \vert, \lvert, \rvert, \Arrowvert$\Arrowvert$ not intended for direct use; used internally to create stretchy delimiters ‖ class PUNCT see also: \|, \Vert, \lVert, \rVert \ast$\ast$ asterisk ∗ class BIN \asymp$\asymp$ asymptotic ≍ class REL \atop general command for making a fraction-like structure, but without the horizontal fraction bar { <subformula1> \atop <subformula2> } Creates a fraction-like structure: ‘numerator’ subformula1 ’denominator’ subformula2 There are separate local groups for subformula1 and subformula2 ; if these local groups are not explicit, then unexpected results may occur, as illustrated in the choose discussion. Examples:  a \atop b yields$a \atop b$a+1 \atop b+2 yields$a+1 \atop b+2${a+1 \atop b+2}+c yields${a+1 \atop b+2}+c$see also: \above, \abovewithdelims, \atopwithdelims, \cfrac, \dfrac, \frac, \genfrac, \over, \overwithdelims \atopwithdelims general command for making a fraction-like structure, but without the horizontal fraction bar; specifies left and right enclosing delimiters { <subformula1> \atopwithdelims <delim1> <delim2> <subformula2> } Creates a fraction-like structure: ‘numerator’ subformula1 ‘denominator’ subformula2 delim1 is put before the structure delim2 is put after the structure For an empty delimiter, use ‘.’ in place of the delimiter. There are separate local groups for subformula1 and subformula2 ; if these local groups are not explicit, then unexpected results may occur, as illustrated in the choose discussion. Examples:  a \atopwithdelims [ ] b yields$a \atopwithdelims [ ] b$a+1 \atopwithdelims . | b+2 yields$a+1 \atopwithdelims . | b+2${a+1 \atopwithdelims \{ \} b+2}+c yields${a+1 \atopwithdelims \{ \} b+2}+c$see also: \above, \abovewithdelims, \atop, \cfrac, \dfrac, \frac, \genfrac, \over, \overwithdelims B  \backepsilon AMSsymbols$\backepsilon$ ∍ class REL  \backprime AMSsymbols$\backprime$ see also: \prime ‵ class ORD  \backsim AMSsymbols$\backsim$ ∽ class REL  \backsimeq AMSsymbols$\backsimeq$ ⋍ class REL \backslash$\backslash$ see also: \setminus ∖ \bar$\bar{}$ bar accent (non-stretchy) ˉ \bar #1 Usually, #1 is a single letter; otherwise, bar is centered over argument. Examples:  \bar x yields$\bar x$\bar X yields$\bar X$\bar xy yields$\bar xy$\bar{xy} yields$\bar{xy}$ \barwedge AMSsymbols$\barwedge$ ⊼ class BIN \Bbb  blackboard-bold for uppercase letters and lowercase ‘k’; if lowercase blackboard-bold letters are not available, then they are typeset in a roman font class ORD \Bbb #1 Whether lower-case letters are displayed in blackboard-bold, or not, depends on the fonts being used. The MathJax web-based fonts don't have lowercase blackboard-bold, but the STIX fonts do; so users with the STIX fonts installed will be able to display lowercase blackboard-bold letters. Examples:  \Bbb R yields$\Bbb R$\Bbb ZR yields$\Bbb ZR$\Bbb{AaBbKk}Cc yields$\Bbb{AaBbKk}Cc$\Bbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\Bbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$see also: \mathbb  \Bbbk AMSsymbols$\Bbbk$ blackboard-bold lowercase k k class ORD  \because AMSsymbols$\because$ ∵ class REL \begin used in \begin{xxx} ... \end{xxx} environments \beta$\beta$ lowercase Greek letter beta β class ORD  \beth AMSsymbols$\beth$ Hebrew letter beth ℶ class ORD  \between AMSsymbols$\between$ ≬ class REL \bf  turns on boldface; affects uppercase and lowercase letters, and digits class ORD {\bf ... } Examples:  \bf AaBb\alpha\beta123 yields$\bf AaBb\alpha\beta123${\bf A B} A B yields${\bf A B} A B$\bf AB \rm CD yields$\bf AB \rm CD$\bf{AB}CD yields$\bf{AB}CD$see also: \mathbf, \boldsymbol \Bigg \bigg \Big \big used to obtain various-sized delimiters; may be followed by any of these Variable-Sized Delimiters Examples: $\Bigg[\bigg[\Big[\big[[$\Bigg[ \bigg[ \Big[ \big[ [ 2.470 em 2.047 em 1.623 em 1.2 em \Biggl \Biggm \Biggr \biggl \biggm \biggr \Bigl \Bigm \Bigl \bigl \bigm \bigr Used to obtain various-sized delimiters, with a left/right/middle context; may be followed by any of these Variable-Sized Delimiters. The ‘l’ (left), ’m’ (middle), and ‘r’ (right) specifications may make reading the source code more meaningful, especially when there are delimiters inside delimiters. Whereas (say) \Bigg produces results of class ORD, we have: • \Biggl produces results of class OPEN • \Biggr produces results of class CLOSE • \Biggm produces results of class REL The spacing for these differ (but may not always be apparent, as it depends on the class of what is next to it). For example, $x\big| y$ ($\,x\big| y\,$) has less space than $x\bigm| y$ ($\,x\bigm| y\,$). Therefore, these commands affect typeset results in a fundamental way; it is best to use the form appropriate for the position of the desired delimiter. \bigcap$\bigcap$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⋂ class OP \bigcirc$\bigcirc$ ◯ class BIN \bigcup$\bigcup$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⋃ class OP $\bigodot\bigoplus\bigotimes$ all change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⨀ class OP ⨁ class OP ⨂ class OP \bigsqcup$\bigsqcup$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⨆ class OP  \bigstar AMSsymbols$\bigstar$ ★ class ORD \bigtriangledown$\bigtriangledown$ ▽ class BIN \bigtriangleup$\bigtriangleup$ △ class REL \biguplus$\biguplus$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⨄ class OP \bigvee$\bigvee$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⋁ class OP \bigwedge$\bigwedge$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ⋀ class OP  \binom AMSmath notation commonly used for binomial coefficients \binom #1 #2 Examples:  \binom n k yields (inline mode)$\binom nk$\binom n k yields (display mode)$\displaystyle\binom nk$\binom{n-1}k-1 yields$\binom{n-1}k-1$\binom{n-1}{k-1} yields$\binom{n-1}{k-1}$see also: \binom, \choose, \dbinom, \tbinom  \blacklozenge AMSsymbols$\blacklozenge$ ⧫ class ORD  \blacksquare AMSsymbols$\blacksquare$ ■ class ORD  \blacktriangle \blacktriangledown both AMSsymbols $\blacktriangle\blacktriangledown$ ▲ class ORD ▼ class ORD  \blacktriangleleft \blacktriangleright both AMSsymbols $\blacktriangleleft\blacktriangleright$ ◀ class BIN ▶ class BIN \bmod$\bmod$ properly spaced as a binary operator class BIN \boldsymbol  as opposed to \bf and \mathbf , \boldsymbol applies to nearly all symbols, not just letters and numbers class ORD \boldsymbol #1 Examples:  \boldsymbol aa yields$\boldsymbol aa$\boldsymbol \alpha\alpha yields$\boldsymbol \alpha\alpha$\boldsymbol{a\alpha}a\alpha yields$\boldsymbol{a\alpha}a\alpha$\boldsymbol{a+2+\alpha+\frac{x+3}{\beta+4}} yields$\boldsymbol{a+2+\alpha+\frac{x+3}{\beta+4}}$\mathbf{a+2+\alpha+\frac{x+3}{\beta+4}} yields$\mathbf{a+2+\alpha+\frac{x+3}{\beta+4}}$see also: \bf, \mathbf \bot$\bot$ ⊥ class ORD \bowtie$\bowtie$ ⋈ class REL  \Box AMSsymbols$\Box$ □ class ORD  \boxdot AMSsymbols$\boxdot$ ⊡ class BIN  \boxed AMSmath puts a box around argument; argument is in math mode \boxed #1 Examples:  \boxed ab yields$\boxed ab$\boxed{ab} yields$\boxed{ab}$\boxed{ab\strut} yields$\boxed{ab\strut}$\boxed{\text{boxed text}} yields$\boxed{\text{boxed text}}$see also: \fbox  \boxminus AMSsymbols \boxplus AMSsymbols \boxtimes AMSsymbols $\boxminus\boxplus\boxtimes$ ⊟ class BIN ⊞ class BIN ⊠ class BIN \brace creates a braced structure { <subformula1> \brace <subformula2> } Examples:  \brace yields$\brace$a\brace b yields$a\brace b$a+b+c\brace d+e+f yields$a+b+c\brace d+e+f$a+{b+c\brace d+e}+f yields$a+{b+c\brace d+e}+f$\bracevert  not intended for direct use; used internally to create stretchy delimiters ⎪ class ORD \brack creates a bracketed structure { <subformula1> \brack <subformula2> } Examples:  \brack yields$\brack$a\brack b yields$a\brack b$a+b+c\brack d+e+f yields$a+b+c\brack d+e+f$a+{b+c\brack d+e}+f yields$a+{b+c\brack d+e}+f$\breve$\breve{}$ breve accent ˘ \breve #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \breve e yields$\breve e$\breve E yields$\breve E$\breve eu yields$\breve eu$\breve{eu} yields$\breve{eu}$\buildrel ... \over ... \buildrel <subformula1> \over #1 The result is of class REL (binary relation), so it has the spacing of a relation. Examples:  \buildrel \alpha\beta \over \longrightarrow yields$\buildrel \alpha\beta \over \longrightarrow$\buildrel \rm def \over {:=} yields$\buildrel \rm def \over {:=}$\bullet$\bullet$ ∙ class BIN  \Bumpeq AMSsymbols \bumpeq AMSsymbols $\Bumpeq\bumpeq$ ≎ class REL ≏ class REL C \cal class ORD turns on calligraphic mode; only affects uppercase letters and digits {\cal ... } Examples:  \cal ABCDEFGHIJKLMNOPQRSTUVWXYZ yields$\cal ABCDEFGHIJKLMNOPQRSTUVWXYZ$\cal 0123456789 yields$\cal 0123456789$\cal abcdefghijklmnopqrstuvwxyz yields$\cal abcdefghijklmnopqrstuvwxyz$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz${\cal AB}AB yields${\cal AB}AB$\cal AB \rm AB yields$\cal AB \rm AB$\cal{AB}CD yields$\cal{AB}CD$see also: \oldstyle, \mathcal \cancel Used to ‘cancel’ (strikeout). \cancel #1 \bcancel #1 Examples:  \frac{(x+1)\cancel{(x+2)}}{3\cancel{(x+2)}} yields$\frac{(x+1)\cancel{(x+2)}}{3\cancel{(x+2)}}$\frac{\bcancel{\frac13}}{\bcancel{\frac13}} = 1 yields$\frac{\bcancel{\frac13}}{\bcancel{\frac13}} = 1$ \Cap AMSsymbols$\Cap$&#x22D2; class BIN see also: \bigcap, \cap, \Cup, \cup, \doublecap, \doublecup \cap$\cap$&#x2229; class BIN see also: \bigcap, \Cap, \Cup, \cup, \doublecap, \doublecup \cases class OPEN for piecewise-defined functions \cases{ [itex] & [itex] \cr <repeat as needed> } a double-backslash can be used in place of \cr ; the final \\ or \cr is optional In$\,\rm\TeX\,$, the second column is automatically in text-mode, while in MathJax it is in math-mode. This behavior will be changed to be consistent with$\,\rm\TeX\,$in a future release of MathJax. Example:  |x| = \cases{ x & \text{if } x\ge 0\cr -x & \text{if } x\lt 0 }  yields$|x| = \cases{ x & \text{if } x\ge 0\cr -x & \text{if } x\lt 0 } $\cdot$\cdot$&#x22C5; class BIN centered dot Examples:  a\cdot b yields$a\cdot b$a\cdotp b yields$a\cdotp b$a\centerdot b yields$a\centerdot b$see also: \cdotp, \cdots, \centerdot \cdotp$\cdotp$&#x22C5; class PUNCT centered dot, punctuation symbol Examples:  \rm s \cdot h yields$\rm s \cdot h$\rm s \cdotp h yields$\rm s \cdotp h$see also: \cdot, \centerdot \cdots$\cdots$&#x22EF; class INNER centered dots; dot dot dot Example: x_1 + \cdots + x_n yields$x_1 + \cdots + x_n$see also: \dots, \ldots  \centerdot AMSsymbols$\centerdot$&#x22C5; class BIN centered dot Examples:  a\cdot b yields$a\cdot b$a\cdotp b yields$a\cdotp b$a\centerdot b yields$a\centerdot b$see also: \cdot, \cdotp  \cfrac AMSmath use for continued fractions \cfrac #1 #2 Examples:  \frac{2}{1+\frac{2}{1+\frac{2}{1}}} yields$\frac{2}{1+\frac{2}{1+\frac{2}{1}}}$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}} yields$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \dfrac, \frac, \genfrac, \over, \overwithdelims \check$\check{}$&#x02C7; check accent \check #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \check o yields$\check o$\check O yields$\check O$\check oe yields$\check oe$\check{oe} yields$\check{oe}$ \checkmark AMSsymbols$\checkmark$#x2713; class ORD \chi$\chi$&#x03C7; class ORD lowercase Greek letter chi \choose notation commonly used for binomial coefficients; different versions for inline and display modes { <subformula1> \choose <subformula2> } There are separate local groups for subformula1 and subformula2 ; if these local groups are not explicit, then unexpected results may occur, as illustrated next. Examples (showing the math delimiters):  $\displaystyle n+1 \choose k+2 $ yields$\displaystyle n+1 \choose k+2$Without an explicit braced group, the local group for subformula1 extends back to the opening math delimiter. That is, this code is interpreted as (color added for emphasis): ${\displaystyle n+1}\choose{k+2}$ Now it is clear that only the n+1 is affected by the \displaystyle switch. $\displaystyle {n+1 \choose k+2} $ yields$\displaystyle {n+1 \choose k+2}$Here, an explicit braced group is used for the \choose command, making both subformulas clear—and the expected result is obtained. Note that it may appear that \displaystyle is taking an argument, but this is not the case: instead, \displaystyle acts as a switch which turns on display mode, and the entire choose command is affected. Examples (showing math delimiters):  $n+1 \choose k+2$ yields$n+1 \choose k+2$$$n+1 \choose k+2$$ yields $$n+1 \choose k+2$$ $1+{n \choose 2}+k$ yields$1+{n \choose 2}+k$see also: \binom, \dbinom, \tbinom \circ$\circ$&#x2218; class BIN Examples:  (f\circ g)(x) = f(g(x)) yields$(f\circ g)(x) = f(g(x))$45^\circ yields$45^\circ$ \circeq AMSsymbols$\circeq$&#x2257; class REL  \circlearrowleft AMSsymbols \circlearrowright AMSsymbols $\circlearrowleft\circlearrowright$ ↺ counterclockwise class REL ↻ clockwise class REL  \circledast AMSsymbols \circledcirc AMSsymbols \circleddash AMSsymbols $\circledast\circledcirc\circleddash$ ⊛ circled asterisk class BIN ⊚ circled circle class BIN ⊝ circled dash class BIN  \circledR AMSsymbols \circledS AMSsymbols $\circledR\circledS$ ® circled R class ORD Ⓢ circled S class ORD  \class [HTML] non-standard; extension is loaded automatically when used; used to specify a CSS class for styling mathematics \class #1 #2 where: • #1 is a CSS class name (without quotes) • #2 is the mathematics to be styled Example: Suppose this CSS style information is provided outside of math mode: <style type="text/css"> .smHighlightRed { font-size:small; background-color:yellow; color:red; } </style>  Then,  ab\class{smHighlightRed}{cdef}gh yields$ab\class{smHighlightRed}{cdef}gh$\clubsuit$\clubsuit$&#x2663; class ORD see also: \diamondsuit, \heartsuit, \spadesuit \colon$\colon$&#x003A; class PUNCT a colon, treated as a punctuation mark (instead of a relation) Examples:  f:A\to B yields$f:A\to B$f\colon A\to B yields$f\colon A\to B$\color  used to specify a color in mathematics \color #1 #2 where: #1 is the desired color #2 is the mathematics to be colored This works differently from standard$\,\rm\LaTeX\,$(where \color is a switch). In a future version of MathJax, it will be possible to load an extension to make the command behave like the$\,\rm\LaTeX\,$version. Examples:  \color{red}{ \frac{1+\sqrt{5}}{2} } yields$\color{red}{ \frac{1+\sqrt{5}}{2} }$\color{#0000FF}AB yields$\color{#0000FF}AB$ \complement AMSsymbols$\complement$&#x2201; class ORD \cong$\cong$&#x2245; class REL congruent see also: \ncong \coprod$\coprod$&#x2210; class OP coproduct \cos$\cos$class OP cosine; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples Examples:  \cos x yields$\cos x$\cos(2x-1) yields$\cos(2x-1)$see also: \sin \cosh$\cosh$class OP hyperbolic cosine; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples hyperbolic cosine Examples:  \cosh x yields$\cosh x$\cosh(2x-1) yields$\cosh(2x-1)$see also: \sinh \cot$\cot$class OP cotangent; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples Examples:  \cot x yields$\cot x$\cot(2x-1) yields$\cot(2x-1)$see also: \tan \coth$\coth$class OP hyperbolic cotangent; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples Examples:  \coth x yields$\coth x$\coth(2x-1) yields$\coth(2x-1)\cr carriage return; line separator in alignment modes and environments in MathJax, these are essentially the same: \\, \newline \csc\csc$class OP cosecant does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples Examples:  \csc x yields$\csc x$\csc(2x-1) yields$\csc(2x-1)$see also: \sec  \cssId [HTML] non-standard; class ORD; extension is loaded automatically when used; used to set a MathML element's ID attribute, so it can be accessed dynamically (e.g., to add an event handler, add CSS styling, or set display status) \cssId #1 #2 where: • #1 is an ID attribute (without quotes) • #2 is the mathematics to be identified by the ID Example: Suppose this HTML and Javascript is provided outside of math mode: <button type="button" onclick="turnRed();"> Click button to turn something red </button> <script type="text/javascript"> function turnRed() { document.getElementById('testID').style.color = "red"; } </script>  Suppose further that the following MathJax code is provided: $$abc \cssId{testID}{def\text{ Something will turn red! }ghi} jkl$$  Then, this HTML/Javascript/MathJax produces: $$abc\cssId{testID}{def\text{ Something will turn red! }ghi}jkl$$ A more meaningful example (with well-commented source code) is provided by Design Science, Inc., and shows how you can display the steps in a proof one line at a time.  \Cup AMSsymbols$\Cup$&#x22D3; class BIN see also: \bigcup, \Cap, \cap, \cup, \doublecap, \doublecup \cup$\cup$&#x222A; class BIN see also: \bigcup, \Cap, \cap, \Cup, \doublecap, \doublecup  \curlyeqprec AMSsymbols \curlyeqsucc AMSsymbols $\curlyeqprec\curlyeqsucc$ ⋞ class REL ⋟ class REL  \curlyvee AMSsymbols \curlywedge AMSsymbols $\curlyvee\curlywedge$ ⋎ class BIN ⋏ class BIN  \curvearrowleft AMSsymbols \curvearrowright AMSsymbols $\curvearrowleft\curvearrowright$ ↶ counterclockwise class REL ↷ clockwise class REL D $\dagger\ddagger$ † dagger class BIN ‡ double dagger class BIN  \daleth AMSsymbols$\daleth$&#x2138; class ORD Hebrew letter daleth  \dashleftarrow AMSsymbols \dashrightarrow AMSsymbols $\dashleftarrow\dashrightarrow$ ⇠ dashed left arrow; non-stretchy class REL ⇢ dashed right arrow; non-stretchy class REL \dashv$\dashv$&#x22A3; class REL  \dbinom AMSmath notation commonly used for binomial coefficients; display version (in both inline and display modes) \dbinom #1 #2 Examples:  \dbinom n k yields (inline mode)$\dbinom n k$\dbinom n k yields (display mode)$\displaystyle\dbinom n k$\dbinom{n-1}k-1 yields$\dbinom{n-1}k-1$\dbinom{n-1}{k-1} yields$\dbinom{n-1}{k-1}$see also: \binom, \choose, \tbinom  \dot \ddot \dddot AMSmath \ddddot AMSmath $\dot{}\ddot{}\dddot{}\ddddot{}$ ˙ dot accent ¨ double dot accent triple dot accent quadruple dot accent \dot #1 \ddot #1 \dddot #1 \ddddot #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \dot x yields$\dot x$\ddot x yields$\ddot x$\dddot x yields$\dddot x$\ddddot x yields$\ddddot x$\ddot x(t) yields$\ddot x(t)$\ddddot{y(x)} yields$\ddddot{y(x)}$\ddots$\ddots$&#x22F1; class INNER three diagonal dots  \DeclareMathOperator AMSmath Multi-letter operator names (like$\,\log\,$,$\,\sin\,$, and$\,\lim\,$) are traditionally typeset in a roman font. \DeclareMathOperator allows you to define your own operator names; they are subsequently typeset using the proper font and spacing; you can control the way that limits appear (see examples below) \DeclareMathOperator #1 #2 where: • #1 is the operator name, including the preceding backslash; only letters a–z and A–Z are allowed; in particular, no numbers are allowed in operator names • #2 is the replacement text for the operator name A named operator is available in any mathematics that appears after it is defined on the page. Examples:  myOp(x) yields$myOp(x)$poor style; the function name should appear in a roman font \text{myOp}(x) yields$\text{myOp}(x)$better; a nuisance to type if used frequently \DeclareMathOperator {\myOp}{myOp} \myOp(x) yields$\DeclareMathOperator {\myOp}{myOp} \myOp(x)$best; once an operator is declared, it can be used in any subsequent mathematics \myOp_a^b(x) yields (inline mode)$\myOp_a^b(x)$standard subscript and superscript position for inline mode \myOp_a^b(x) yields (display mode)$\displaystyle\myOp_a^b(x)$standard subscript and superscript position for display mode \DeclareMathOperator* {\myOP}{myOP} \myOP_a^b(x) yields (inline mode)$\DeclareMathOperator* {\myOP}{myOP} \myOP_a^b(x)$operator names are case-sensitive, so \myOp is different from \myOP ; if displaystyle limits are desired in both inline and display modes, then use DeclareMathOperator* instead of DeclareMathOperator \def for defining your own commands (control sequences, macros, definitions); must appear (within math delimiters) before it is used; alternatively, you can define macros using the MathJax configuration options in the <head> \def\myCommandName{ <replacement text> } Example: \def\myHearts{\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} \myHearts\myHearts  yields:$ \def\myHearts{\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} \myHearts\myHearts $A definition may take one or more arguments: Example: \def\myHearts#1#2{\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} \myHearts{red}{blue}  yields:$ \def\myHearts#1#2{\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} \myHearts{red}{blue} $see also: \newcommand \deg$\deg$class OP degree; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples $\Delta\delta$ Δ uppercase Greek letter delta class ORD δ lowercase Greek letter delta class ORD see also: \varDelta \det$\det$class OP determinant; does not change size; default limit placement can be changed using \limits and \nolimits; does not change size; see the Big Operators Table for more examples Examples:  \det_{\rm sub} yields (inline mode)$\det_{\rm sub}$\det_{\rm sub} yields (display mode)$\displaystyle\det_{\rm sub}$\det\limits_{\rm sub} yields (inline mode)$\det\limits_{\rm sub}$\det\nolimits_{\rm sub} yields (display mode)$\displaystyle\det\nolimits_{\rm sub}$ \dfrac AMSmath fractions; display version (in both inline and display modes) \dfrac #1 #2 Examples:  \dfrac a b yields (inline mode)$\dfrac a b$\dfrac a b yields (display mode)$\displaystyle\dfrac a b$\frac a b yields (inline mode)$\frac a b$\dfrac{a-1}b-1 yields$\dfrac{a-1}b-1$\dfrac{a-1}{b-1} yields$\dfrac{a-1}{b-1}$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \cfrac, \frac, \genfrac, \over, \overwithdelims  \diagdown AMSsymbols \diagup AMSsymbols $\diagdown\diagup$ ╲ diagonal down (from left to right) class ORD ╱ diagonal up (from left to right) class ORD  \Diamond AMSsymbols \diamond $\Diamond\diamond$ ◊ large diamond class ORD ⋄ small diamond class BIN \diamondsuit$\diamondsuit$&#x2662; class ORD see also: \clubsuit, \heartsuit, \spadesuit  \digamma AMSsymbols$\digamma$&#x03DD; class ORD \dim$\dimclass OP dimension; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples \displaylines to display any number of centered formulas (without any alignment) \displaylines{ [itex] \cr <repeat as needed> } a double-backslash can be used in place of the \cr; the final \\ or \cr is optional Example:  \displaylines{ a = a\\ \text{if } a=b \text{ then } b=a\\ \text{if } a=b \text{ and } b=c \text{ then } a=c }  yields \displaylines{ a = a\\ \text{if } a=b \text{ then } b=a\\ \text{if } a=b \text{ and } b=c \text{ then } a=c } $see also: gather \displaystyle class ORD used to over-ride automatic style rules and force display style; stays in force until the end of math mode or the braced group, or until another style is selected { \displaystyle ... } Example: In inline mode:  \frac ab+\displaystyle\frac ab+\textstyle\frac ab +\scriptstyle\frac ab+\scriptscriptstyle\frac ab yields:$\frac ab + \displaystyle\frac ab+\textstyle\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab$Example: In inline mode: \frac ab + {\displaystyle \frac cd + \frac ef} + \frac gh yields$\frac ab + {\displaystyle \frac cd + \frac ef} + \frac gh$Example: In inline mode: \frac ab + \displaystyle{\frac cd + \frac ef} + \frac gh yields$\frac ab + \displaystyle{\frac cd + \frac ef} + \frac gh$see also: \textstyle, \scriptstyle, \scriptscriptstyle \div$\div$&#x00F7; class BIN division symbol  \divideontimes AMSsymbols$\divideontimes$&#x22C7; class BIN  \Doteq AMSsymbols \doteq $\Doteq\doteq$ ≑ class REL ≐ class REL  \dotplus AMSsymbols$\dotplus$&#x2214; class BIN \dots$\dots$&#x2026; class INNER lower dots; ellipsis; ellipses; dot dot dot In$\,\rm\LaTeX\,$, \dots chooses either \cdots or \ldots depending on the context; MathJax, however, always gives lower dots. Examples:  x_1, \dots, x_n yields$x_1, \dots, x_n$x_1 + \dots + x_n yields$x_1 + \dots + x_n$x_1 + \dotsb + x_n yields$x_1 + \dotsb + x_n$x_1 + \cdots + x_n yields$x_1 + \cdots + x_n$see also: \cdots, \ldots, \dotsb, \dotsc, \dotsi, \dotsm, \dotso  ⋯ \dotsb class INNER dots with binary operations and relations$x_1 + x_2 +\dotsb + x_n$… \dotsc class INNER dots with commas$x_1,x_2,\dotsc,x_n$⋯ \dotsi class INNER dots with integrals$\int_{A_1}\int_{A_2}\dotsi\int_{A_n}$⋯ \dotsm class INNER dots with multiplication$x_1x_2\dotsm x_n$… \dotso class INNER other dots$A_1\dotso A_n$see also: \cdots, \dots, \ldots  \doublebarwedge AMSsymbols$\doublebarwedge$&#x2A5E; BIN  \doublecap AMSsymbols \doublecup AMSsymbols $\doublecap\doublecup$ ⋒ class BIN ⋓ class BIN see also: \Cap, \Cup, \cap, \cup $\downarrow\Downarrow$ ↓ down arrow; non-stretchy class REL ⇓ double down arrow; non-stretchy class REL  \downdownarrows AMSsymbols$\downdownarrows$&#x21CA; class REL down down arrows; non-stretchy  \downharpoonleft AMSsymbols \downharpoonright AMSsymbols $\downharpoonleft\downharpoonright$ ⇃ down harpoon left; non-stretchy class REL ⇂ down harpoon right; non-stretchy class REL see also: \leftharpoondown, \leftharpoonup E \ell$\ell$&#x2113; class ORD \emptyset$\emptyset$&#x2205; class ORD empty set see also: \varnothing \end used in \begin{xxx} ... \end{xxx} environments \enspace \enspace is a 0.5em space Example:  |\enspace|\enspace| yields$|\enspace|\enspace|$\epsilon$\epsilon&#x03F5; class ORD lowercase Greek letter epsilon see also: \varepsilon \eqalign equation alignment; for aligning multi-line displays at a single place \eqalign{ [itex] & [itex] \cr <repeat as needed> } the ampersand is placed where alignment is desired; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional; supports only a single \tag, which is vertically centered Example: \eqalign{ 3x - 4y &= 5\cr x + 7 &= -2y }  yields: \eqalign{ 3x - 4y &= 5\cr x + 7 &= -2y } Example: A [itex] component may be empty: \eqalign{ (a+b)^2 &= (a+b)(a+b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 }  yields: \eqalign{ (a+b)^2 &= (a+b)(a+b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 } Example: The result of \eqalign is a vertically-centered block; you can use more than one in the same display: \left\{ \eqalign{ a &= 1\\ b &= 2\\ c &= 3 }\right\} \qquad \eqalign{ ax + by &= c \\ x + 2y &= 3 }  yields: \left\{ \eqalign{ a &= 1\\ b &= 2\\ c &= 3 }\right\} \qquad \eqalign{ ax + by &= c \\ x + 2y &= 3 } see also: \eqalignno, the align environment, \tag \eqalignno equation alignment with optionally numbered (tagged) lines \eqalignno{ [itex] & [itex] & <equation tag> \cr <repeat as needed> } the first ampersand is placed where alignment is desired; the second ampersand is used just before a tag; if there is no tag, then the final & <equation tag> is omitted; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional Example: \eqalignno{ 3x - 4y &= 5 &(\dagger) \cr x + 7 &= -2y &(\ddagger)\cr z &= 2 }  yields: \eqalignno{ 3x - 4y &= 5 &(\dagger)\cr x + 7 &= -2y &(\ddagger)\cr z &= 2 } see also: \eqalign, \leqalignno, the align environment  \eqcirc AMSsymbols\eqcirc$&#x2256; class REL  \eqsim AMSsymbols$\eqsim$&#x2242; class REL  \eqslantgtr AMSsymbols \eqslantless AMSsymbols $\eqslantgtr\eqslantless$ &##x2A96; class REL &##x2A95; class REL \equiv$\equiv$&#x2261; class REL Error Messages; page processing log When you're working with a MathJax page, you may want to see the log of messages generated during page processing (particularly if something has gone wrong). To do this, type javascript:alert(MathJax.Message.Log())  in the browser's location URL box, and then refresh the page. If the alert box is too big to see the close button, just press ‘enter’ to close the alert box. \eta$\eta$&#x03B7; class ORD lowercase Greek letter eta  \eth AMSsymbols$\eth$&#x00F0; class ORD \exists$\exists$&#x2203; class ORD there exists see also: \nexists \exp$\exp$class OP exponential function; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples F  \fallingdotseq AMSsymbols$\fallingdotseq$&#x2252; class REL falling dot sequence; see also: \risingdotseq \fbox puts a box around argument; argument is in text mode equivalent to: \boxed{\text{#1}} \fbox #1 where #1 is rendered as text Examples:  \boxed{Hi there!} yields$\boxed{Hi there!}$\fbox{Hi there!} yields$\fbox{Hi there!}$see also: \boxed  \Finv AMSsymbols$\Finv$&#x2132; class ORD \flat$\flat$&#x266D; class ORD musical flat symbol see also: \natural, \sharp \forall$\forall$&#x2200; class ORD universal quantifier; for all; for every; for each  \frac AMSmath fractions; displays differently in inline and display modes \frac #1 #2 Examples:  \frac a b yields (inline mode)$\frac a b$\frac a b yields (display mode)$\displaystyle\frac a b$\frac{a-1}b-1 yields$\frac{a-1}b-1$\frac{a-1}{b-1} yields$\frac{a-1}{b-1}$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \cfrac, \dfrac, \genfrac, \over, \overwithdelims \frak class ORD turns on fraktur; affects uppercase and lowercase letters, and digits {\frak ... } Examples:  \frak ABCDEFGHIJKLMNOPQRSTUVWXYZ yields$\frak ABCDEFGHIJKLMNOPQRSTUVWXYZ$\frak 0123456789 yields$\frak 0123456789$\frak abcdefghijklmnopqrstuvwxyz yields$\frak abcdefghijklmnopqrstuvwxyz${\frak AB}AB yields${\frak AB}AB$\frak AB \rm AB yields$\frak AB \rm AB${\frak AB \cal AB} AB yields${\frak AB \cal AB} AB$see also: \mathfrak \frown$\frown$&#x2322; class REL see also: \smallfrown, \smallsmile, \smile G  \Game AMSsymbols$\Game$&#x2141; class ORD \Gamma$\Gamma$&#x0393; class ORD uppercase Greek letter gamma see also: \varGamma \gamma$\gamma$&#x03B3; class ORD lowercase Greek letter gamma \gcd$\gcd$class OP greatest common divisor; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples Examples:  \gcd_{\rm sub}^{\rm sup} yields (inline mode)$\gcd_{\rm sub}^{\rm sup}$\gcd_{\rm sub}^{\rm sup} yields (display mode)$\displaystyle\gcd_{\rm sub}^{\rm sup}$ \ge \geq \geqq AMSsymbols \geqslant AMSsymbols $\ge\geq\geqq\geqslant$ ≥ \ge ≥ \geq ≧ \geqq ⩾ \geqslant all class REL greater than or equal to see also: \ngeq, \ngeqq, \ngeqslant  \genfrac AMSmath the most general command for defining fractions with optional delimiters, line thickness, and specified style \genfrac #1 #2 #3 #4 #5 #6 where: • #1 is the left delimiter (empty, for no left delimiter) • #2 is the right delimiter (empty, for no right delimiter) • #3 is the fraction bar thickness (set to 0pt to make it disappear) • #4 is either 0, 1, 2, or 3, where: • 0 denotes \displaystyle • 1 denotes \textstyle • 2 denotes \scriptstyle • 3 denotes \scriptscriptstyle • #5 is the numerator • #6 is the denominator Example:  \genfrac(]{0pt}{2}{a+b}{c+d} yields$\genfrac(]{0pt}{2}{a+b}{c+d}$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \cfrac, \dfrac, \frac, \over, \overwithdelims \gets$\gets$&#x2190; class REL left arrow; non-stretchy \gg$\gg$&#x226B; class REL  \ggg AMSsymbols \gggtr AMSsymbols $\ggg\gggtr$ ⋙ class REL ⋙ class REL  \gimel AMSsymbols$\gimel$&#x2137; class ORD Hebrew letter gimel  \gtrapprox AMSsymbols \gnapprox AMSsymbols $\gtrapprox\gnapprox$ ⪆ class REL ⪊ class REL  \gneq AMSsymbols \gneqq AMSsymbols \gvertneqq AMSsymbols $\gneq\gneqq\gvertneqq$ ⪈ class REL ≩ class REL ≩ class REL  \gtrsim AMSsymbols \gnsim AMSsymbols $\gtrsim\gnsim$ ≳ class REL ⋧ class REL \grave$\grave{}$&#x02CB; grave accent \grave #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \grave e yields$\grave e$\grave E yields$\grave E$\grave eu yields$\grave eu$\grave{eu} yields$\grave{eu}$\gt$\gt$&#x003E; class REL greater than see also: \ngtr  \gtrdot AMSsymbols$\gtrdot$&#x22D7; class REL  \gtreqless AMSsymbols \gtreqqless AMSsymbols $\gtreqless\gtreqqless$ ⋛ class REL ⪌ class REL  \gtrless AMSsymbols$\gtrless$&#x2277; class REL H \hat$\hat{}$&#x02CA; non-stretchy hat accent \hat #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \hat\imath yields$\hat\imath$\hat\jmath yields$\hat\jmath$\hat ab yields$\hat ab$\hat{ab} yields$\hat{ab}$see also: \widehat \hbar$\hbar$&#x210F; class ORD Planck's constant \hbox class ORD horizontal box; contents are treated as text, but you can switch to math mode inside; text appears in \rm \hbox #1 Examples:  \hbox{\alpha a }\alpha a yields$\hbox{\alpha a }\alpha a$\hbox{This is a sentence.} yields$\hbox{This is a sentence.}$\hbox{for all$x > 0$} yields$\hbox{for all $x > 0$}$in MathJax, these are essentially the same: \text, \mbox see also: \rm works in many of the environments to create a horizontal line (\hline), or a horizontal dashed line (\hdashline) Putting \hdashline or \hline first or last encases the entire structure (which is different from standard$\,\rm\LaTeX\,$behavior):  \begin{matrix} \hdashline x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \end{matrix}  yields$ \begin{matrix} \hdashline x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \end{matrix} $\begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \\ \hline \end{matrix}  yields$ \begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \\ \hline \end{matrix} $Putting \hdashline or \hline at the beginning of any subsequent row puts a line over that row:  \begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ \hline x_{31} & x_{32} \end{matrix}  yields$ \begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ \hline x_{31} & x_{32} \end{matrix} $You can combine effects, and put in struts (as desired) for additional vertical spacing:  \begin{matrix} \hline x_{11} & x_{12} \\ x_{21} & x_{22} \strut \\ \hdashline x_{31} & x_{32} \strut \end{matrix}  yields$ \begin{matrix} \hline x_{11} & x_{12} \\ x_{21} & x_{22} \strut \\ \hdashline x_{31} & x_{32} \strut \end{matrix} $\heartsuit$\heartsuit&#x2661; class ORD see also: \clubsuit, \diamondsuit, \spadesuit \hfil \hfill horizontal glue; horizontal fill (added in MathJax 2.5); can be used to set horizontal alignment in matrices and arrays (as in old-fashioned\,\TeX\,$layout); it ‘expands’ to fill available horizontal space, pushing contents on right or left to the boundary Example: \begin{matrix} xxxxxx & xxxxxx & xxxxxx \cr ab & \hfil ab & ab\hfil\cr \end{matrix}  yields$ \begin{matrix} xxxxxx & xxxxxx & xxxxxx \cr ab & \hfil ab & ab\hfil \cr \end{matrix} $see also: \hskip, \hspace, \kern, \mkern, \mskip, \mspace \hom$\hom$class OP homomorphism; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples $\hookleftarrow\hookrightarrow$ ↩ non-stretchy ↪ non-stretchy both class REL \hphantom class ORD horizontal phantom Sometimes you want to pretend that something is there, for spacing reasons, but you don't want it to appear—you want it to be invisible—you want it to be a phantom. The box created by \hphantom has the width of its argument, but its height and depth are zero (so it doesn't contribute to any vertical spacing issues). In other words, \hphantom creates horizontal space equal to that produced by its argument, but doesn't create any vertical space. \hphantom #1 Example: \begin{array}{l} \text{Side Angle Side}\\ \text{S}\hphantom{\text{ide }}\text{A}\hphantom{\text{ngle }}\text{S} \end{array}  yields$ \begin{array}{l} \text{Side Angle Side}\\ \text{S}\hphantom{\text{ide }}\text{A}\hphantom{\text{ngle }}\text{S} \end{array} $see also: \phantom, \vphantom \href used to make a math object into a link \href{ <url> } #1 where the argument (#1) is the clickable area Example:  \href{http://www.onemathematicalcat.org}{M^{A^{T^H}}} yields$\href{http://www.onemathematicalcat.org}{M^{A^{T^H}}}$\hskip horizontal glue; horizontal space; horizontal skipping; Example: w\hskip1em i\hskip2em d\hskip3em e\hskip4em r  yields$ w\hskip1em i\hskip2em d\hskip3em e\hskip4em r $in MathJax, these all behave the same: \hspace, \kern, \mkern, \mskip, \mspace  \hslash AMSsymbols$\hslash$&#x210F; class ORD perhaps an alternative form of Planck's constant \hspace horizontal glue; horizontal space; horizontal skipping Example: s\hspace7ex k\hspace6ex i\hspace5ex n\hspace4ex n\hspace3ex i\hspace2ex e\hspace1ex r  yields$ s\hspace7ex k\hspace6ex i\hspace5ex n\hspace4ex n\hspace3ex i\hspace2ex e\hspace1ex r $in MathJax, these all behave the same: \hskip, \kern, \mkern, \mskip, \mspace \Huge \huge both class ORD turns on huge mode and an even bigger Huge mode {\Huge ... } {\huge ... } Examples:  \huge AaBb\alpha\beta123\frac ab\sqrt x yields$\huge AaBb\alpha\beta123\frac ab\sqrt x${\huge A B} A B yields${\huge A B} A B$A\alpha\huge A\alpha \Huge A\alpha yields$A\alpha\huge A\alpha \Huge A\alpha$see also: \LARGE, \Large, \large I \iddots$\def\iddots{{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu \raise12mu{.}}}\iddots$Not in MathJax Library inner diagonal dots; This macro must be supplied by the user, if desired. Davide Cervone provided the code (given here) in the MathJax User Group. To use this macro, put the following definition in either inline or display mathematics: $
\def\iddots{
{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}}
$ Then, in any subsequent mathematics:  \iddots yields$\iddots$Instead of providing the definition inside math delimiters in the body, you can add the definition to your configuration using the Macros property of the TeX block: <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { Macros: { iddots: "{\\kern3mu\\raise1mu{.}\\kern3mu\\raise6mu{.}\\kern3mu\\raise12mu{.}}" }}}); </script>   \idotsint AMSmath$\idotsint$class OP changes size; can change limit placement using \limits; see the Big Operators Table for examples \iff$\iff$&#x27FA; with a thick space on both sides if and only if; is equivalent to; non-stretchy Example: A\iff B yields$A\iff B$ \iiiint AMSmath \iiint \iint \int $\iiiint\iiint\iint\int$ four occurrences of ∫ ∭ ∬ ∫ all class OP; see the Big Operators Table for examples Compare the different limit placements (both in display mode):  \int_a^b yields $$\int_a^b$$ \intop_a^b yields $$\intop_a^b$$ see also: \intop $\intop$&#x222B; (with movable limits) class OP See the Big Operators Table for examples. see also: \iiiint, \iiint, \iint, \int \Im$\Im$&#x2111; class ORD \imath$\imath$&#x0131; class ORD a dotless ‘i’; better to use when accented Examples:  \hat i yields$\hat i$\hat\imath yields$\hat\imath$see also: \jmath  \impliedby AMSsymbols$\impliedby$&#x27F8; with a thick space on both sides non-stretchy Example: P\impliedby Q yields$P\impliedby Q$ \implies AMSsymbols$\implies$&#x27F9; with a thick space on both sides non-stretchy Example: P\implies Q yields$P\implies Q$\in$\in$&#x2208; class REL is in; is an element of; indicates membership in a set; see also: \ni, \notin, \owns \inf$\inf$class OP infimum; least upper bound; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples Examples:  \inf_{\rm limit} yields (inline mode)$\inf_{\rm limit}$\inf_{\rm limit} yields (display mode)$\displaystyle\inf_{\rm limit}$see also: \sup \infty$\infty$&#x221E; class ORD infinity  \injlim AMSmath$\injlim$class OP injective limit; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples see also: \varinjlim  \intercal AMSsymbols$\intercal$&#x22BA; class BIN \iota$\iota$&#x03B9; class ORD lowercase Greek letter iota \it class ORD turns on math italic mode; to return to math italic mode if it had been turned off {\it ... } Examples:  {\bf ab \it ab} ab yields${\bf ab \it ab} ab$\rm for\ all\ {\it x}\ in\ \Bbb R yields$\rm for\ all\ {\it x}\ in\ \Bbb R$\Delta\Gamma\Lambda{\it \Delta\Gamma\Lambda} yields$\Delta\Gamma\Lambda{\it \Delta\Gamma\Lambda}$see also: \mathit, \mit J \jmath$\jmath$&#x0237; class ORD a dotless ‘j’; better to use when accented Examples:  \hat j yields$\hat j$\hat\jmath yields$\hat\jmath$see also: \imath  \Join AMSsymbols$\Join$&#x22C8; class REL K \kappa$\kappa$&#x03BA; class ORD lowercase Greek letter kappa see also: \varkappa \ker$\ker$class OP kernel; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples \kern to get a specified amount of horizontal space; a negative argument forces ‘backing up’, so items can overlap Examples:  |\kern 2ex|\kern 2em|\kern 2pt| yields$|\kern 2ex|\kern 2em|\kern 2pt|$\rm I\kern-2.5pt R yields$\rm I\kern-2.5pt R$in MathJax, these all behave the same: \hskip, \hspace, \mkern, \mskip, \mspace L $\Lambda\lambda$ uppercase Greek letter lambda Λ class ORD lowercase Greek letter lambda λ class ORD see also: \varLambda \land$\land$ logical AND ∧ class BIN see also: \lor, \wedge \langle$\langle$ left angle bracket; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⟨ class OPEN Example:  \left\langle \matrix{a & b\cr c & d} \right\rangle  yields$\left\langle \matrix{a & b\cr c & d} \right\rangle$see also: \rangle  turns on large typestyles; affects all math all class ORD {\LARGE ... } {\Large ... } {\large ... } Examples:  \Large AaBb\alpha\beta123\frac ab yields$\Large AaBb\alpha\beta123\frac ab${\Large A B} A B yields${\Large A B} A B$AB \large AB \Large AB \LARGE AB yields$AB \large AB \Large AB \LARGE AB$\Large{AB}CD yields$\Large{AB}CD$see also: \huge, \Huge \LaTeX$\LaTeX$ the LaTeX logo class ORD Example: \rm\LaTeX yields$\rm\LaTeX$see also: \TeX \lbrace$\lbrace$ left brace: non-stretchy when used alone; stretchy when used with \left or \right (see below) class OPEN Examples:  \lbrace \frac ab, c \rbrace yields$\lbrace \frac ab, c \rbrace$\left\lbrace \frac ab, c \right\rbrace yields$\left\lbrace \frac ab, c \right\rbrace$see also: \rbrace, \{ \} \lbrack$\lbrack$ left bracket: non-stretchy when used alone; stretchy when used with \left or \right (see below); class OPEN Examples:  \lbrack \frac ab, c \rbrack yields$\lbrack \frac ab, c \rbrack$\left\lbrack \frac ab, c \right\rbrack yields$\left\lbrack \frac ab, c \right\rbrack$see also: \rbrack, [ ] \lceil$\lceil$ left ceiling; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⌈ class OPEN Example:  \left\lceil \matrix{a & b\cr c & d} \right\rceil  yields$\left\lceil \matrix{a & b\cr c & d} \right\rceil$see also: \rceil, \lfloor, \rfloor \ldotp$\ldotp$ lower dot, punctuation symbol . class PUNCT Examples:  \rm s \ldotp h yields$\rm s \ldotp h$\rm s.h yields$\rm s.h$see also: \cdotp \ldots$\ldots$ lower dots; ellipsis; ellipses; dot dot dot … class INNER Example: x_1,\ldots,x_n yields$x_1,\ldots,x_n$see also: \cdots, \dots  \le \leq \leqq AMSsymbols \leqslant AMSsymbols $\le\leq\leqq\leqslant$ less than or equal to ≤ class REL less than or equal to ≤ class REL less than or equal to ≦ class REL less than or equal to ⩽ class REL see also: \nleq, \nleqq, \nleqslant  \leadsto AMSsymbols$\leadsto$ ⇝ class REL \left  used for stretchy delimiters; see the Variable-Sized Delimiters Table for details Examples:  \left( \frac12 \right) yields$\left( \frac12 \right)$\left\updownarrow \phantom{\frac12} \right\Updownarrow yields$\left\updownarrow \phantom{\frac12} \right\Updownarrow$see also: \right $\leftarrow\Leftarrow$ left arrow; non-stretchy ← class REL left arrow; non-stretchy ⇐ class REL see also: \nleftarrow, \nLeftarrow  \leftarrowtail AMSsymbols$\leftarrowtail$ left arrow tail; non-stretchy ↢ class REL see also: \rightarrowtail $\leftharpoondown\leftharpoonup$ left harpoon arrow; non-stretchy ↽ class REL left harpoon arrow; non-stretchy ↼ class REL  \leftleftarrows AMSsymbols$\leftleftarrows$ left left arrows; non-stretchy ⇇ class REL $\leftrightarrow\Leftrightarrow$ left right arrow; non-stretchy ↔ class REL left right arrow; non-stretchy ⇔ class REL see also: \nleftrightarrow, \nLeftrightarrow  \leftrightarrows AMSsymbols$\leftrightarrows$ left right arrows; non-stretchy ⇆ class REL  \leftrightharpoons AMSsymbols$\leftrightharpoons$ left right harpoons; non-stretchy ⇋ class REL  \leftrightsquigarrow AMSsymbols$\leftrightsquigarrow$ left right squiqqle arrow; non-stretchy ↭ class REL \leftroot used to fine-tune the placement of the index inside \sqrt or \root (see examples) \sqrt[... \leftroot #1 ...]{...} \root ... \leftroot #1 ... \of {...} where the argument is a small integer: a positive integer moves the index to the left; a negative integer moves the index to the right Examples:  \sqrt[3]{x} yields$\sqrt[3]{x}$\sqrt[3\leftroot1]{x} yields$\sqrt[3\leftroot1]{x}$\root 3 \of x yields$\root 3 \of x$\root 3\leftroot{-1} \of x yields$\root 3\leftroot{-1} \of x$\root 3\leftroot{-1}\uproot2 \of x yields$\root 3\leftroot{-1}\uproot2 \of x$see also: \uproot, \root  \leftthreetimes AMSsymbols$\leftthreetimes ⋋ class BIN \leqalignno equation alignment with optionally numbered (tagged) lines; in\rm\TeX, \leqalignno puts the tags on the left, but MathJax doesn't implement this behavior; currently, tags appear in a column on the right separated from the equations by a fixed amount of space (so they don't work like tags in the AMS math environments); this may be fixed in a future version of MathJax \leqalignno{ [itex] & [itex] & <equation tag> \cr <repeat as needed> } the first ampersand is placed where alignment is desired; the second ampersand is used just before a tag; if there is no tag, then the final & <equation tag> is omitted; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional; output is the same in both inline and display modes (except for the amount of vertical space before and after); Example: \leqalignno{ 3x - 4y &= 5 &(\dagger) \cr x + 7 &= -2y &(\ddagger)\cr z &= 2 }  yields: \leqalignno{ 3x - 4y &= 5 &(\dagger) \cr x + 7 &= -2y &(\ddagger)\cr z &= 2 } see also: \eqalignno; the align environment  \lessapprox AMSsymbols\lessapprox$ see also: \lnapprox ⪅ class REL  \lessdot AMSsymbols$\lessdot$ ⋖ class REL  \lesseqgtr AMSsymbols \lesseqqgtr AMSsymbols $\lesseqgtr\lesseqqgtr$ ⋚ class REL ⪋ class REL  \lessgtr AMSsymbols$\lessgtr$ ≶ class REL  \lesssim AMSsymbols$\lesssim$ see also: \lnsim ≲ class REL \lfloor$\lfloor$ left floor; non-stretchy when used alone; stretchy when used with \left or \right ⌊ class OPEN see also: \rfloor, \lceil, \rceil \lg$\lg$ does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples class OP \lgroup$\lgroup$ left group; non-stretchy when used alone; stretchy when used with \left or \right ⟮ class OPEN Example:  \left\lgroup \matrix{a & b\cr c & d} \right\rgroup  yields$\left\lgroup \matrix{a & b\cr c & d} \right\rgroup$see also: \rgroup  \lhd AMSsymbols$\lhd$ left-hand diamond ⊲ class REL see also: \rhd \lim$\lim$ limit; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \lim_{n\rightarrow\infty} f(x) = \ell (inline mode) yields$\lim_{n\rightarrow\infty} f(x) = \ell$\lim_{n\rightarrow\infty} f(x) = \ell (display mode) yields $$\lim_{n\rightarrow\infty} f(x) = \ell$$ \liminf$\liminf$ limit inferior; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \liminf_{n\rightarrow\infty} x_n = \ell (inline mode) yields$\liminf_{n\rightarrow\infty} x_n = \ell$\liminf_{n\rightarrow\infty}\ x_n = \ell (display mode) yields $$\liminf_{n\rightarrow\infty}\ x_n = \ell$$ see also: \varliminf \limits used to set limits above/below any token of class OP; see the Big Operators table for more information and examples Examples:  \int_a^b f(x)\,dx (inline mode) yields$\int_a^b f(x)\,dx$\int\limits_a^b f(x)\,dx (inline mode) yields$\int\limits_a^b f(x)\,dx$\int_a^b f(x)\,dx (display mode) yields $$\int_a^b f(x)\,dx$$ \int\limits_a^b f(x)\,dx (display mode) yields $$\int\limits_a^b f(x)\,dx$$ \mathop{x}\limits_0^1 yields$\mathop{x}\limits_0^1$see also: \nolimits \limsup$\limsup$ limit superior; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \limsup_{n\rightarrow\infty} x_n (inline mode) yields$\limsup_{n\rightarrow\infty} x_n$\limsup_{n\rightarrow\infty}\ x_n (display mode) yields $$\limsup_{n\rightarrow\infty}\ x_n$$ see also: \varlimsup \ll$\ll$ &x226A; class REL \llap  left overlap class ORD \llap #1 creates a box of width zero; the argument is then placed just to the left of this zero-width box (and hence will overlap whatever lies to the left); proper use of \llap and \rlap in math expressions is somewhat delicate Examples:  a\mathrel{{=}\llap{/}}b yields$a\mathrel{{=}\llap{/}}b${=} forces the equal to not have REL spacing (since it is not adjacent to ORD's) and \mathrel{} forces the compound symbol (equal with overlapping slash) to be treated as a single REL a\mathrel{{=}\llap{/\,}}b yields$a\mathrel{{=}\llap{/\,}}b$the thinspace ‘\,’ improves the spacing a=\mathrel{\llap{/\,}}b yields$a=\mathrel{\llap{/\,}}b$this works because the spacing between adjacent REL's is zero see also: \rlap  \llcorner AMSsymbols \lrcorner AMSsymbols $\llcorner\lrcorner$ lower left corner └ class REL lower right corner ┘ class REL These are technically delimiters, but MathJax doesn't stretch them like it should. see also: \ulcorner, \urcorner  \Lleftarrow AMSsymbols$\Lleftarrow$ non-stretchy ⇚ class REL  \lll AMSsymbols \llless AMSsymbols $\lll\llless$ ⋘ class REL ⋘ class REL \lmoustache$\lmoustache$ left moustache; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⎰ class OPEN Example:  \left\lmoustache \phantom{\matrix{a & b\cr c & d}} \right\rmoustache  yields $$\left\lmoustache \phantom{\matrix{a & b\cr c & d}} \right\rmoustache$$ see also: \rmoustache \ln$\ln$ natural logarithm; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples class OP  \lnapprox AMSsymbols$\lnapprox$ see also: \lessapprox ⪉ class REL  \lneq AMSsymbols \lneqq AMSsymbols $\lneq\lneqq$ see also: \leq ⪇ class REL see also: \leqq ≨ class REL \lnot$\lnot$ logical not ¬ class ORD see also: \neg  \lnsim AMSsymbols$\lnsim$ see also: \lesssim ⋦ class REL \log$\log$ logarithm; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for examples class OP $\longleftarrow\Longleftarrow\longrightarrow\Longrightarrow$ non-stretchy ⟵ class REL non-stretchy ⟸ class REL non-stretchy ⟶ class REL non-stretchy ⟹ class REL $\longleftrightarrow\Longleftrightarrow$ non-stretchy ⟷ class REL non-stretchy ⟺ class REL \longmapsto$\longmapsto$ long maps to ⟼ class REL see also: \mapsto  \looparrowleft AMSsymbols \looparrowright AMSsymbols $\looparrowleft\looparrowright$ non-stretchy ↫ class REL non-stretchy ↬ class REL \lor$\lor$ logical OR ∨ class BIN see also: \land, \vee \lower lowers the argument by the amount specified in <dimen>; in actual$\rm\TeX$, the argument to \lower (and \raise ) must be an \hbox , but in MathJax it can be any expression (using an \hbox is allowed, but not required) Example:  l\lower 2pt {owe} r yields$l\lower 2pt {owe} r$see also: \raise  \lozenge AMSsymbols$\lozenge$ ◊ class ORD  \Lsh AMSsymbols$\Lsh$ left shift; non-stretchy ↰ class REL see also: \Rsh \lt$\lt$ less than < class REL see also: \nless  \ltimes AMSsymbols$\ltimes$ see also: \rtimes ⋉ class BIN  \lvert AMSmath \lVert AMSmath $\lvert\lVert$ both non-stretchy when used alone; ∣ class OPEN stretchy when used with \left or \right ∥ class OPEN Example:  \left\lvert\frac{\frac ab}{\frac cd}\right\rvert yields$\left\lvert\frac{\frac ab}{\frac cd}\right\rvert$see also: \rvert, \rVert, |, \|  \lvertneqq AMSsymbols$\lvertneqq$ ≨ class REL M  \maltese AMSsymbols$\maltese$ ✠ class ORD \mapsto$\mapsto$ maps to; non-stretchy math operator ↦ class REL see also: \longmapsto \mathbb  blackboard-bold for uppercase letters and lowercase ‘k’; if lowercase blackboard-bold letters are not available, then they are typeset in a roman font class ORD \mathbb #1 Whether lower-case letters are displayed in blackboard-bold, or not, depends on the fonts being used. The MathJax web-based fonts don't have lowercase blackboard-bold, but the STIX fonts do; so users with the STIX fonts installed will be able to display lowercase blackboard-bold letters. Examples:  \mathbb R yields$\mathbb R$\mathbb ZR yields$\mathbb ZR$\mathbb{AaBbKk}Cc yields$\mathbb{AaBbKk}Cc$\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$see also: \Bbb \mathbf  boldface for uppercase and lowercase letters and digits class ORD \mathbf #1 Examples:  \mathbf{AaBb\alpha\beta123} yields$\mathbf{AaBb\alpha\beta123}$\mathbf ZR yields$\mathbf ZR$\mathbf{uvw}xyz yields$\mathbf{uvw}xyz$see also: \bf, \boldsymbol \mathbin  gives the correct spacing to make an object into a binary operator; binary operators have some extra space around them; creates an element of class BIN class BIN \mathbin #1 Examples:  a\text{op} b yields$a\text{op} b$a\mathbin{\text{op}} b yields$a\mathbin{\text{op}} b$a\Diamond b yields$a\Diamond b$a\mathbin{\Diamond}b yields$a\mathbin{\Diamond} b$\mathcal  calligraphic font for uppercase letters and digits class ORD \mathcal #1 Examples:  \mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\mathcal{0123456789} yields$\mathcal{0123456789}$\mathcal{abcdefghijklmnopqrstuvwxyz} yields$\mathcal{abcdefghijklmnopqrstuvwxyz}$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz$\mathcal{AB}AB yields$\mathcal{AB}AB$see also: \cal, \oldstyle \mathchoice provides content that is dependent on the current style (display, text, script, or scriptscript); can be used in defining a macro for general use \mathchoice #1 #2 #3 #4 where: • #1 is rendered when the \mathchoice appears in display style • #2 is rendered when the \mathchoice appears in text style • #3 is rendered when the \mathchoice appears in script style • #4 is rendered when the \mathchoice appears in scriptscript style Examples:  \mathchoice{D}{T}{S}{SS} (in display style) yields $$\mathchoice{D}{T}{S}{SS}$$ \mathchoice{D}{T}{S}{SS} (in text style) yields$\mathchoice{D}{T}{S}{SS}$\mathchoice{D}{T}{S}{SS} (in script style) yields$\scriptstyle\mathchoice{D}{T}{S}{SS}$\mathchoice{D}{T}{S}{SS} (in scriptscript style) yields$\scriptscriptstyle\mathchoice{D}{T}{S}{SS}$Here's a nice example from the$\rm\TeX$Book: Define: \def\puzzle{\mathchoice{D}{T}{S}{SS}}  Then:  \puzzle{\puzzle\over\puzzle^{\puzzle^\puzzle}} yields (in display mode) $$\def\puzzle{\mathchoice{D}{T}{S}{SS}} \puzzle{\puzzle\over\puzzle^{\puzzle^\puzzle}}$$ \puzzle{\puzzle\over\puzzle^{\puzzle^\puzzle}} yields (in inline mode)$\puzzle{\puzzle\over\puzzle^{\puzzle^\puzzle}}$\mathclose  forces the argument to be treated in the ‘closing’ class; for example, like ‘$)$’ and ‘$]$’; creates an element of class CLOSE class CLOSE \mathclose #1 Examples:  a + \lt b\gt + c yields$a + \lt b\gt + c$a + \mathopen\lt b\mathclose\gt + c yields$a + \mathopen\lt b\mathclose\gt + c$see also: \mathopen \mathfrak  fraktur font for uppercase and lowercase letters and digits (and a few other characters) class ORD \mathfrak #1 Examples:  \mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\mathfrak{0123456789} yields$\mathfrak{0123456789}$\mathfrak{abcdefghijklmnopqrstuvwxyz} yields$\mathfrak{abcdefghijklmnopqrstuvwxyz}$\mathfrak{AB}AB yields$\mathfrak{AB}AB$see also: \frak \mathinner  some constructions are meant to appear ‘inside’ other formulas, and should be surrounded by additional space in certain circumstances; this classification is forced on the argument by using \mathinner class INNER \mathinner #1 Examples:  ab\text{inside}cd yields$ab\text{inside}cd$ab\mathinner{\text{inside}}cd yields$ab\mathinner{\text{inside}}cd$\mathit  math italic mode class ORD \mathit #1 Examples:  \rm abc \mathit{def} ghi yields$\rm abc \mathit{def} ghi$in MathJax, this is the same as: \mit and \it \mathop  forces the argument to be treated in the ‘large operator’ class; for example, like ‘$\sum$’; creates an element of class OP class OP \mathop #1 Examples:  atbtc yields$atbtc$a\mathop{t}b\mathop{t}c yields$a\mathop{t}b\mathop{t}c$\star_a^b yields (in display mode) $$\star_a^b$$ \mathop{\star}_a^b yields (in display mode) $$\mathop{\star}_a^b$$ \mathopen  forces the argument to be treated in the ‘opening’ class; for example, like ‘$($’ and ‘$[$’; creates an element of class OPEN class OPEN \mathopen #1 Examples:  a + \lt b\gt + c yields$a + \lt b\gt + c$a + \mathopen\lt b\mathclose\gt + c yields$a + \mathopen\lt b\mathclose\gt + c$see also: \mathclose \mathord  forces the argument to be treated in the ‘ordinary’ class; for example, like ‘$/$’; spacing is determined by pairs of tokens; there is no extra spacing between adjacent ORD's (as in the second example below); there is extra spacing between an ORD and a BIN (as in the first example below); creates an element of class ORD class ORD \mathord #1 Examples:  a+b+c yields$a+b+c$a\mathord{+}b\mathord{+}c yields$a\mathord{+}b\mathord{+}c$1,234,567 yields$1,234,567$1\mathord{,}234{,}567 yields$1\mathord{,}234{,}567$\mathpunct  forces the argument to be treated in the ‘punctuation’ class; for example, like ‘$,$’; punctuation tends to have some extra space after the symbol; returns an element of class PUNCT class PUNCT \mathpunct #1 Examples:  1.234 yields$1.234$1\mathpunct{.}234 yields$1\mathpunct{.}234$\mathrel  forces the argument to be treated in the ‘relation’ class; for example, like ‘$=$’ and ‘$\gt$’; relations have a bit more space on both sides than binary operators; returns an element of class REL class REL \mathrel #1 Examples:  a \# b yields$ a \# b$a \mathrel{\#} b yields$ a \mathrel{\#} b$ \mathring AMSmath$\mathring{}$ ˚ \mathring #1 Examples:  \mathring A yields$\mathring A$\mathring{AB}C yields$\mathring{AB}C$\mathrm  roman typestyle for uppercase and lowercase letters class ORD \mathrm #1 Examples:  \mathrm{AaBb\alpha\beta123} yields$\mathrm{AaBb\alpha\beta123}$\mathrm ZR yields$\mathrm ZR$\mathrm{uvw}xyz yields$\mathrm{uvw}xyz$see also: \rm \mathscr  script typestyle for uppercase letters; if lowercase script letters are not available, then they are typeset in a roman typestyle class ORD \mathscr #1 Whether lower-case letters are displayed in script, or not, depends on the fonts being used. The MathJax web-based fonts don't have lowercase script, but the STIX fonts do; so users with the STIX fonts installed will be able to display lowercase script letters. Examples:  \mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\mathscr{0123456789} yields$\mathscr{0123456789}$\mathscr{abcdefghijklmnopqrstuvwxyz} yields$\mathscr{abcdefghijklmnopqrstuvwxyz}$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz$\mathscr{AB}AB yields$\mathscr{AB}AB$see also: \scr \mathsf  sans serif typestyle for uppercase and lowercase letters and digits; also affects uppercase greek (as do the other font switches, like \rm, \it, \bf, \mathrm, \mathit, \mathbf, etc). class ORD \mathsf #1 Examples:  \mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\mathsf{0123456789} yields$\mathsf{0123456789}$\mathsf{abcdefghijklmnopqrstuvwxyz} yields$\mathsf{abcdefghijklmnopqrstuvwxyz}$\Delta\Gamma\Lambda\mathsf{\Delta\Gamma\Lambda} yields$\Delta\Gamma\Lambda\mathsf{\Delta\Gamma\Lambda}$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz$\mathsf{AB}AB yields$\mathsf{AB}AB$see also: \sf \mathstrut  an invisible box whose width is zero; its height and depth are the same as a parenthesis ‘$($’; can be used to achieve more uniform appearance in adjacent formulas class ORD Examples:  \sqrt3 + \sqrt\alpha yields$\sqrt3 + \sqrt\alpha$\sqrt{\mathstrut 3} + \sqrt{\mathstrut\alpha} yields$\sqrt{\mathstrut 3} + \sqrt{\mathstrut\alpha}$\mathtt  typewriter typestyle for uppercase and lowercase letters and digits; also affects uppercase Greek class ORD \mathtt #1 Examples:  \mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ} yields$\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\mathtt{0123456789} yields$\mathtt{0123456789}$\mathtt{abcdefghijklmnopqrstuvwxyz} yields$\mathtt{abcdefghijklmnopqrstuvwxyz}$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz$\Delta\Gamma\Lambda\mathtt{\Delta\Gamma\Lambda} yields$\Delta\Gamma\Lambda\mathtt{\Delta\Gamma\Lambda}$\mathtt{AB}AB yields$\mathtt{AB}ABsee also: \tt \matrix matrix (without any delimiters) \matrix{ [itex] & [itex] ... \cr <repeat as needed> } alignment occurs at the ampersands; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional Example:  \matrix{ a & b \cr c & d }  yields \matrix{ a & b \cr c & d } $see also: \array \max$\max$ maximum; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \max_{\rm sub} yields (inline mode)$\max_{\rm sub}$\max_{\rm sub} yields (display mode)$\displaystyle\max_{\rm sub}$see also: \min \mbox  creates a box just wide enough to hold the text in its argument; no linebreaks are allowed in the text; text appears in \rm class ORD \mbox <text argument> Examples:  a + b \mbox{ (are you paying attention?) } = c yields$a + b \mbox{ (are you paying attention?) } = c$a + b \text{ (are you paying attention?) } = c yields$a + b \text{ (are you paying attention?) } = c$in MathJax, these are essentially the same: \text, \hbox see also: \rm  \measuredangle AMSsymbols$\measuredangle$ ∡ class ORD  \mho AMSsymbols$\mho$ ℧ class ORD \mid$\mid$ the spacing is perfect for use in set-builder notation ∣ class REL Examples:  \{x | x\gt 1\} yields$\{x | x\gt 1\}$\{x \mid x\gt 1\} yields$\{x \mid x\gt 1\}$see also: \nmid, \shortmid, \nshortmid \min$\min$ minimum; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \min_{\rm sub} yields (inline mode)$\min_{\rm sub}$\min_{\rm sub} yields (display mode)$\displaystyle\min_{\rm sub}$see also: \max \mit  math italic typestyle class ORD \mit #1 Examples:  \mit{\Gamma\Delta\Theta\Omega} yields$\mit{\Gamma\Delta\Theta\Omega}$\mathit{\Gamma\Delta\Theta\Omega} yields$\mathit{\Gamma\Delta\Theta\Omega}$\Gamma\Delta\Theta\Omega yields$\Gamma\Delta\Theta\Omega$in MathJax, this is the same as: \mathit and \it \mkern gives horizontal space Examples:  ab yields$ab$a\mkern18mu b yields$a\mkern18mu b$a\mkern18pt b yields$a\mkern18pt b$in MathJax, these all behave the same: \hskip, \hspace, \kern, \mskip, \mspace \mod$\mod{}$modulus operator; modulo; the leading space depends on the style: displaystyle has 18 mu, others 12 mu; 2 thinspaces of following space; for things like equations modulo a number \mod #1 Example:  3\equiv 5 \mod 2 yields$3\equiv 5 \mod 2$see also: \pmod, \bmod \models$\models$ ⊨ class REL shifts boxes to the left or right \moveleft <dimen> <box> \moveright <dimen> <box> In actual$\rm\TeX$, these require an \hbox (or some box) as an argument, and can only appear in vertical mode; MathJax is less picky: you don't need an actual box, and MathJax doesn't have a vertical mode; these are not really designed as user-level macros, but instead allow existing macros to work; the box takes up its original space (unlike something like \llap or \rlap ), but its contents are shifted (without affecting its bounding box) Examples:  \rm tight yields$\rm tight$\rm t\moveleft3pt ight yields$\rm t\moveleft3pt ight$\rm t\moveleft3pt i\moveleft3pt g\moveleft3pt h\moveleft3pt t yields$\rm t\moveleft3pt i\moveleft3pt g\moveleft3pt h\moveleft3pt t$\rm t\moveleft3pt i\moveleft6pt g\moveleft9pt h\moveleft12pt t yields$\rm t\moveleft3pt i\moveleft6pt g\moveleft9pt h\moveleft12pt t$\square\square\moveleft 2em {\diamond\diamond} yields$\square\square\moveleft 2em {\diamond\diamond}$\square\square\moveright 2em {\diamond\diamond} yields$\square\square\moveright 2em {\diamond\diamond}$see also: \raise, \lower \mp$\mp$ minus plus ∓ class BIN see also: \pm \mskip gives horizontal space Examples:  ab yields$ab$a\mskip18mu b yields$a\mskip18mu b$a\mskip18pt b yields$a\mskip18pt b$in MathJax, these all behave the same: \hskip, \hspace, \kern, \mkern, \mspace \mspace gives horizontal space Examples:  ab yields$ab$a\mspace18mu b yields$a\mspace18mu b$a\mspace18pt b yields$a\mspace18pt b$in MathJax, these all behave the same: \hskip, \hspace, \kern, \mkern, \mskip \mu$\mu$ lowercase Greek letter mu μ class ORD  \multimap AMSsymbols$\multimap$ ⊸ class REL N \nabla$\nabla$ ∇ class ORD \natural$\natural$ see also: \flat, \sharp ♮ class ORD  \ncong AMSsymbols$\ncong$ not congruent ≆ class REL see also: \cong \ne$\ne$ not equal ≠ class REL see also: equals, \neq \nearrow$\nearrow$ northeast arrow; non-stretchy ↗ class REL see also: \nwarrow, \searrow, \swarrow \neg$\neg$ negate; negation ¬ class ORD see also: \lnot  \negthinspace AMSmath \negmedspace AMSmath \negthickspace AMSmath  negative thin space negative medium space negative thick space Examples:  ab yields$ab$a\negthinspace b yields$a\negthinspace b$a\negmedspace b yields$a\negmedspace b$a\negthickspace b yields$a\negthickspace b$see also: \thinspace \neq$\neq$ see also: equals, \ne ≠ class REL \newcommand for defining your own commands (control sequences, macros, definitions); \newcommand must appear (within math delimiters) before it is used; if desired, you can use the TeX.Macros property of the configuration to define macros in the head \newcommand\myCommandName [ <optional # of arguments, from 1 to 9> ] { <replacement text> }  The bracketed # of arguments is omitted when there are no arguments. Example (no arguments): \newcommand\myHearts {\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} \myHearts\myHearts  yields:$ \newcommand\myHearts {\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} \myHearts\myHearts $A definition may take one or more arguments: Example (two arguments): \newcommand\myHearts[2] {\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} \myHearts{red}{blue}  yields:$ \newcommand\myHearts[2] {\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} \myHearts{red}{blue} $see also: \def, \newenvironment \newenvironment for defining your own environments; \newenvironment must appear (within math delimiters) before it is used \newenvironment{myEnvironmentName} [ <optional # of arguments, from 1 to 9> ] { <replacement text for each occurrence of \begin{myEnvironmentName}> } { <replacement text for each occurrence of \end{myEnvironmentName}> }  The bracketed # of arguments is omitted when there are no arguments. There must not be a command having the same name as the environment: for example, to use \begin{myHeart}...\end{myHeart} there may not be a command \myHeart. Example (no arguments): \newenvironment{myHeartEnv} {\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} {\text{ forever}} \begin{myHeartEnv} \end{myHeartEnv}  yields:$ \newenvironment{myHeartEnv} {\color{purple}{\heartsuit}\kern-2.5pt\color{green}{\heartsuit}} {\text{ forever}} \begin{myHeartEnv} \end{myHeartEnv} $An environment may take one or more arguments: Example (two arguments): \newenvironment{myHeartEnv}[2] {\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} {\text{ forever}} \begin{myHeartEnv}{red}{blue} \end{myHeartEnv}  yields:$ \newenvironment{myHeartEnv}[2] {\color{#1}{\heartsuit}\kern-2.5pt\color{#2}{\heartsuit}} {\text{ forever}} \begin{myHeartEnv}{red}{blue} \end{myHeartEnv} see also: \def, \newcommand \newline line separator in alignment modes and environments in MathJax, these are essentially the same: \cr, \\  \nexists AMSsymbols\nexists$ see also: \exists ∄ class ORD  \ngeq AMSsymbols \ngeqq AMSsymbols $\ngeq\ngeqq$ not greater than or equal to ≱ class REL not greater than or equal to ≱ class REL see also: \geq, \geqq  \ngeqslant AMSsymbols$\ngeqslant$ slanted not greater than or equal to ⪈ class REL see also: \geqslant  \ngtr AMSsymbols$\ngtr$ not greater than ≯ class REL see also: \gt \ni$\ni$ backwards ‘in’; contains ∋ class REL see also: \in  \nleftarrow AMSsymbols \nLeftarrow AMSsymbols $\nleftarrow\nLeftarrow$ ↚ class REL ⇍ class REL see also: \leftarrow, \Leftarrow  \nleftrightarrow AMSsymbols \nLeftrightarrow AMSsymbols $\nleftrightarrow\nLeftrightarrow$ ↮ class REL ⇎ class REL see also: \leftrightarrow, \Leftrightarrow  \nleq AMSsymbols \nleqq AMSsymbols $\nleq\nleqq$ not less than or equal to ≰ class REL not less than or equal to ≰ class REL see also: \leq, \leqq  \nleqslant AMSsymbols$\nleqslant$ slanted not less than or equal to ⪇ class REL see also: \leqslant  \nless AMSsymbols$\nless$ see also: \lt ≮ class REL  \nmid AMSsymbols$\nmid$ see also: \mid ∤ class REL  \nobreakspace AMSmath  Example:   class ORD  a\nobreakspace b yields$a\nobreakspace b$in MathJax, this is the same as: \ (backslash space) \nolimits  used to change the default placement of limits; only allowed on items of class OP Examples:  \sum_{k=1}^n a_k yields (in display mode) $$\sum_{k=1}^n a_k$$ \sum\nolimits_{k=1}^n a_k yields (in display mode) $$\sum\nolimits_{k=1}^n a_k$$ see also: \limits \normalsize  turns on normal size class ORD {\normalsize ... } Example:  \rm \scriptsize script \normalsize normal \large large yields$\rm \scriptsize script \normalsize normal \large large $see also: \scriptsize \not$\not{}$ used to negate relations / class REL Examples:  \not\gt yields$\not\gt$\ngtr yields$\ngtr$ \notag AMSmath  used in AMS math environments that do automatic equation numbering, to suppress the equation number; since MathJax doesn't implement auto-numbering (as of version 1.1a), it is basically a no-op, although it will cancel an explicit \tag ; when auto-numbering is added, then this will work as expected; \notag is included now for compatibility with existing TeX code (to prevent throwing an error, even though it has no effect) class ORD \notin$\notin$ see also: \in ∉ class REL  \nparallel AMSsymbols$\nparallel$ not parallel ∦ class REL see also: \parallel  \nprec AMSsymbols$\nprec$ see also: \prec ⊀ class REL  \npreceq AMSsymbols$\npreceq$ see also: \preceq ⋠ class REL  \nrightarrow AMSsymbols \nRightarrow AMSsymbols $\nrightarrow\nRightarrow$ ↛ class REL ⇏ class REL see also: \rightarrow, \Rightarrow  \nshortmid AMSsymbols$\nshortmid$ see also: \mid, \shortmid ∤ class REL  \nshortparallel AMSsymbols$\nshortparallel$ see also: \parallel, \shortparallel ∦ class REL  \nsim AMSsymbols$\nsim$ see also: \sim ≁ class REL  \nsubseteq AMSsymbols \nsubseteqq AMSsymbols $\nsubseteq\nsubseteqq$ ⊈ class REL ⊈ class REL see also: \subseteq, \subseteqq  \nsucc AMSsymbols \nsucceq AMSsymbols $\nsucc\nsucceq$ ⊁ class REL ⋡ class REL see also: \succ, \succeq  \nsupseteq AMSsymbols \nsupseteqq AMSsymbols $\nsupseteq\nsupseteqq$ ⊉ class REL ⊉ class REL see also: \supseteq, \supseteqq  \ntriangleleft AMSsymbols \ntrianglelefteq AMSsymbols $\ntriangleleft\ntrianglelefteq$ ⋪ class REL ⋬ class REL see also: \triangleleft, \trianglelefteq  \ntriangleright AMSsymbols \ntrianglerighteq AMSsymbols $\ntriangleright\ntrianglerighteq$ ⋫ class REL ⋭ class REL see also: \triangleright, \trianglerighteq \nu$\nu$ lowercase Greek letter nu ν class ORD  \nVDash AMSsymbols \nVdash AMSsymbols \nvDash AMSsymbols \nvdash AMSsymbols $\nVDash\nVdash\nvDash\nvdash$ ⊯ class REL ⊮ class REL ⊭ class REL ⊬ class REL see also: \Vdash, \vDash, \vdash \nwarrow$\nwarrow$ northwest arrow; non-stretchy ↖ class REL see also: \nearrow, \searrow, \swarrow O $\odot\ominus\oplus\oslash\otimes$ ⊙ class BIN ⊖ class BIN ⊕ class BIN ⊘ class BIN ⊗ class BIN \oint$\oint$ changes size; can change limit placement using \limits; see the Big Operators Table for examples ∮ class OP \oldstyle  this is intended for oldstyle numbers; it is a switch that turns on oldstyle mode; the way it works in$\rm\TeX$is to select the caligraphic font (which is where the oldstyle numbers are stored), so it has the side effect of selecting caligraphic upper-case letters; MathJax does the same for compatibility class ORD {\oldstyle ... } Examples:  \oldstyle 0123456789 yields$\oldstyle 0123456789$\oldstyle ABCDEFGHIJKLMNOPQRSTUVWXYZ yields$\oldstyle ABCDEFGHIJKLMNOPQRSTUVWXYZ$\oldstyle abcdefghijklmnopqrstuvwxyz yields$\oldstyle abcdefghijklmnopqrstuvwxyz$abcdefghijklmnopqrstuvwxyz yields$abcdefghijklmnopqrstuvwxyz${\oldstyle AB}AB yields${\oldstyle AB}AB$\oldstyle AB \rm AB yields$\oldstyle AB \rm AB$\oldstyle{AB}CD yields$\oldstyle{AB}CD$see also: \cal, \mathcal $\omega\Omega$ lowercase Greek letter omega ω class ORD uppercase Greek letter omega Ω class ORD see also: \varOmega \omicron$\omicron$ lowercase Greek letter omicron ο class ORD  \operatorname AMSmath  This is similar to \DeclareMathOperator, but rather than defining a macro, it produces an instance of an operator like \lim . For example, \operatorname{myOp} is equivalent to the use of \myOp , after having defined \DeclareMathOperator{\myOp}{myOp} If displaystyle limits are desired in both inline and display modes, then use operatorname* instead of operatorname class OP Examples:  \operatorname{myFct}(x) yields$\operatorname{myFct}(x)$\operatorname*{myFct}_a^b(x) yields (in inline mode)$\operatorname*{myFct}_a^b(x)$See \DeclareMathOperator for further explanation and examples. \over general command for making fractions { <subformula1> \over <subformula2> } Creates a fraction: numerator: subformula1 denominator: subformula2 Examples:  a \over b yields$a \over b$a+1 \over b+2 yields$a+1 \over b+2${a+1 \over b+2}+c yields${a+1 \over b+2}+c$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \cfrac, \dfrac, \frac, \genfrac, \overwithdelims \overbrace puts a (stretchy) over-brace over the argument; can use ‘^’ to place an optional superscript over the overbrace; can use ‘_’ to place an optional subscript below the argument \overbrace #1 Example:  \overbrace{x + \cdots + x}^{n\rm\ times}_{\text{(note here)} yields$\overbrace{x + \cdots + x}^{n\rm\ times}_{\text{(note here)}}$see also: \underbrace $\overleftarrow{}\overrightarrow{}\overleftrightarrow{}$ ← stretchy over left arrow → stretchy over right arrow ↔ stretchy over left right arrow \overleftarrow #1 \overrightarrow #1 \overleftrightarrow #1 Examples:  \overleftarrow{\text{the argument}} yields$\overleftarrow{\text{the argument}}$\overrightarrow{AB} yields$\overrightarrow{AB}$\overrightarrow{AB\strut} yields$\overrightarrow{AB\strut}$\overleftrightarrow{\hspace1in} yields$\overleftrightarrow{\hspace1in}$\overline$\overline{}$ stretchy overline ‾ \overline #1 Examples:  \overline{AB} yields$\overline{AB}$\overline a yields$\overline a$\overline{\text{a long argument}} yields$\overline{\text{a long argument}}$\overparen puts a (stretchy) over-parenthesis (over-arc, frown) over the argument (new in MathJax 2.6) \overparen #1 Example:  \overparen a \quad \overparen ab \quad \overparen{ab} \quad \overparen{abc} \quad \overparen{abcdef} \quad \overparen{\underparen{abcd}}  yields$\overparen a \quad \overparen ab \quad \overparen{ab} \quad \overparen{abc} \quad \overparen{abcdef} \quad \overparen{\underparen{abcd}} $see also: \underparen, \smallfrown, \frown, \smallsmile, \smile \overset \overset #1 #2 oversets argument #1 (in scriptstyle) over argument #2 Examples:  \overset{\rm top}{\rm bottom} yields$\overset{\rm top}{\rm bottom}$\overset ab yields $$\overset ab$$ a\,\overset{?}{=}\,b yields $$a\,\overset{?}{=}\,b$$ see also: \atop, \underset \overwithdelims general command for making fractions; uses default thickness for fraction bar for current size specifies left and right enclosing delimiters { <subformula1> \overwithdelims <delim1> <delim2> <subformula2> } Creates a fraction: numerator subformula1 denominator subformula2 delim1 is put before the fraction delim2 is put after the fraction For an empty delimiter, use ‘.’ in place of the delimiter. Examples:  a \overwithdelims [ ] b yields$a \overwithdelims [ ] b$a+1 \overwithdelims . | b+2 yields$a+1 \overwithdelims . | b+2${a+1 \overwithdelims \{ \} b+2}+c yields${a+1 \overwithdelims \{ \} b+2}+c$see also: \above, \abovewithdelims, \atop, \atopwithdelims, \cfrac, \dfrac, \frac, \genfrac, \over \owns$\owns$ see also: \ni, \in ∋ class REL P \parallel$\parallel$ see also: \nparallel ∥ class REL \partial$\partial$Example:  \frac{\partial f}{\partial x} yields$\frac{\partial f}{\partial x}$&#x2202; class ORD \perp$\perp$ perpendicular to ⊥ class REL \phantom  phantom (both horizontal and vertical) class ORD Sometimes you want to pretend that something is there, for spacing reasons, but you don't want it to appear—you want it to be invisible—you want it to be a phantom. The box created by \phantom has width, height and depth equal to its argument. In other words, \phantom creates horizontal and vertical space equal to that of its argument, even though the argument isn't visible. \phantom #1 Examples:  \sqrt{\frac ab} \sqrt{\phantom{\frac ab}}  yields$ \sqrt{\frac ab} \sqrt{\phantom{\frac ab}} $\frac{2x+3y-\phantom{5}z} {\phantom{2}x+\phantom{3}y+5z}  yields$\displaystyle \frac{2x+3y-\phantom{5}z} {\phantom{2}x+\phantom{3}y+5z}$\Gamma^{\phantom{i}j}_{i\phantom{j}k}  yields$\displaystyle \Gamma^{\phantom{i}j}_{i\phantom{j}k} $\matrix{1&-1\cr 2&\phantom{-}3}  yields$\displaystyle \matrix{1&-1\cr 2&\phantom{-}3}$see also: \hphantom, \vphantom $\phi\Phi$ lowercase Greek letter phi &##x03D5; class ORD uppercase Greek letter phi Φ class ORD see also: \varphi, \varPhi $\pi\Pi$ lowercase Greek letter pi π class ORD uppercase Greek letter Pi Π class ORD see also: \varpi, \varPi  \pitchfork AMSsymbols$\pitchfork$ ⋔ class REL \pm$\pm plus or minus &x00B1; class BIN see also: \mp \pmatrix  matrix enclosed in parentheses class OPEN \pmatrix{ [itex] & [itex] ... \cr <repeat as needed> } alignment occurs at the ampersands; a double-backslash can be used in place of the \cr ; the final \\ or \cr is optional Example:  A = \pmatrix{ a_{11} & a_{12} & \ldots & a_{1n} \cr a_{21} & a_{22} & \ldots & a_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{m1} & a_{m2} & \ldots & a_{mn} \cr }  yields A = \pmatrix{ a_{11} & a_{12} & \ldots & a_{1n} \cr a_{21} & a_{22} & \ldots & a_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{m1} & a_{m2} & \ldots & a_{mn} \cr } $see also: \matrix \pmb  poor man's bold; it works by duplicating its argument slightly offset, giving a bold effect (at least in the horizontal direction); doesn't work well for horizontal lines, like$\,-\,$or$\,+\,$class ORD \pmb #1 Examples:  a \pmb a \boldsymbol a yields$a \pmb a \boldsymbol a$\pmb{a+b-c}\ \ a+b-c yields$\pmb{a+b-c}\ \ a+b-c$\pmod$\pmod{}$parenthesized modulus operator; parenthesized modulo; 18 mu of leading space before the opening parenthesis in display style; 8 mu of leading space before the opening parenthesis in other styles; 6 mu of space after the word mod \pmod #1 Examples:  5\equiv 8 \pmod 3 yields$5\equiv 8 \pmod 3$\pmod{n+m} yields$\pmod{n+m}$see also: \mod, \bmod \pod$\pod{}$parenthesized argument with leading space; 18 mu of leading space before the opening parenthesis in display style; 8 mu of leading space before the opening parenthesis in other styles \pod #1 Examples:  x=y\pod{\text{inline mode}} yields$x=y\pod{\text{inline mode}}$x=y\pod{\text{display mode}} yields$\displaystyle x=y\pod{\text{display mode}}$\Pr$\Pr$ does not change size; default limit placement can be changed using \limits and \nolimits; does not change size; see the Big Operators Table for more examples class OP Examples:  \Pr_{\rm sub} yields (inline mode)$\Pr_{\rm sub}$\Pr_{\rm sub} yields (display mode)$\displaystyle\Pr_{\rm sub}$\prec$\prec$ see also: \nprec ≺ class REL  \precapprox AMSsymbols \precnapprox AMSsymbols $\precapprox\precnapprox$ ⪷ class REL ⪹ class REL  \preccurlyeq AMSsymbols$\preccurlyeq$ ≼ class REL  \preceq \precneqq AMSsymbols $\preceq\precneqq$ ⪯ class REL ⪵ class REL see also: \npreceq  \precsim AMSsymbols \precnsim AMSsymbols $\precsim\precnsim$ ≾ class REL ⋨ class REL \prime$\prime$ prime character ′ class ORD Examples:  f' yields$f'$f\prime yields$f\prime$f^\prime yields$f^\prime$f^{\prime\prime} yields$f^{\prime\prime}$f'' yields$f''$see also: \backprime, prime symbol \prod$\prod$ changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for more examples ∏ class OP Examples:  \prod_{j=1}^n yields (in inline mode)$\prod_{j=1}^n$\prod_{j=1}^n yields (in display mode) $$\prod_{j=1}^n$$  \projlim AMSmath$\projlim$ projective limit; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP see also: \varprojlim \propto$\propto$ see also: \varpropto ∝ class REL $\psi\Psi$ lowercase Greek letter psi ω class ORD uppercase Greek letter psi Ω class ORD see also: \varPsi Q$ \def\mark{\rlap{\normalsize\textstyle |}\kern 1px} $\quad \qquad \quad is a 1em space \qquad is a 2em space Examples:  |\quad|\quad| yields$|\quad|\quad|$|\qquad\hphantom{|}| yields$|\qquad\hphantom{|}|$R \raise raises the argument by the amount specified in <dimen>; in actual$\rm\TeX$, the argument to \raise (and \lower ) must be an \hbox , but in MathJax it can be any expression (using an \hbox is allowed, but not required) Example:  h\raise 2pt {ighe} r yields$h\raise 2pt {ighe} r$see also: \lower \rangle$\rangle$ right angle bracket; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⟩ class CLOSE Example:  \left\langle \matrix{a & b\cr c & d} \right\rangle  yields$\left\langle \matrix{a & b\cr c & d} \right\rangle$see also: \langle \rbrace$\rbrace$ right brace; non-stretchy when used alone; stretchy when used with \left or \right (see below) class CLOSE Example:  \left\lbrace \matrix{a & b\cr c & d} \right\rbrace  yields$\left\lbrace \matrix{a & b\cr c & d} \right\rbrace$see also: \lbrace \rbrack$\rbrack$ right bracket; non-stretchy when used alone; stretchy when used with \left or \right (see below) class CLOSE Examples:  \lbrack \frac ab, c \rbrack yields$\lbrack \frac ab, c \rbrack$\left\lbrack \frac ab, c \right\rbrack yields$\left\lbrack \frac ab, c \right\rbrack$see also: \lbrack, [ ] \rceil$\rceil$ right ceiling; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⌉ class CLOSE Example:  \left\lceil \matrix{a & b\cr c & d} \right\rceil  yields$\left\lceil \matrix{a & b\cr c & d} \right\rceil$see also: \lceil, \lfloor, \rfloor \Re$\Re$ ℜ class ORD \renewcommand equivalent to \newcommand; for clarity of code, you may choose to use \renewcommand when re-defining a macro; this is different from actual$\,\rm\TeX\,$, where \renewcommand only allows redefining of an existing command see also: \def, \newcommand, \newenvironment \require (non-standard) This is a MathJax-specific macro that can be used to load MathJax$\rm\TeX$extensions (like the AMSmath extension) from within math mode, rather than having to include it in the configuration. For example, $\require{AMSsymbols}$ would cause MathJax to load the extensions/TeX/AMSsymbols.js file at that point. Since many people use MathJax in blogs and wikis that may not have all the extensions loaded, this makes it possible to load a lesser-used extension on a particular page, without having to include it in every page.  \restriction AMSsymbols$\restriction$ ↾ class REL \rfloor$\rfloor$ right floor; non-stretchy when used alone; stretchy when used with \left or \right ⌋ class CLOSE see also: \lfloor, \lceil, \rceil \rgroup$\rgroup$ right group; non-stretchy when used alone; stretchy when used with \left or \right ⟮ class CLOSE Example:  \left\lgroup \matrix{a & b\cr c & d} \right\rgroup  yields$\left\lgroup \matrix{a & b\cr c & d} \right\rgroup$see also: \lgroup  \rhd AMSsymbols$\rhd$ right-hand diamond ⊳ class REL see also: \lhd \rho$\rho$ lowercase Greek letter rho � class ORD see also: \varrho \right  used for stretchy delimiters; see the Variable-Sized Delimiters Table for details Can be followed by:  delimiter: sample code: yields: ( ) \left( \frac12 \right)$\left( \frac12 \right)$\updownarrow\Updownarrow \left\updownarrow \phantom{\frac12} \right\Updownarrow$\left\updownarrow \phantom{\frac12} \right\Updownarrow$see also: \left $\rightarrow\Rightarrow$ non-stretchy → class REL non-stretchy ⇒ class REL see also: \nrightarrow, \nRightarrow, \to  \rightarrowtail AMSsymbols$\rightarrowtail$ right arrow tail; non-stretchy ↣ class REL see also: \leftarrowtail $\rightharpoondown\rightharpoonup$ non-stretchy ⇁ class REL non-stretchy ⇀ class REL see also: \leftharpoondown, \rightharpoondown  \rightleftarrows AMSsymbols$\rightleftarrows$ right left arrows; non-stretchy ⇄ class REL  \rightleftharpoons AMSsymbols$\rightleftharpoons$ right left harpoons; non-stretchy ⇌ class REL  \rightrightarrows AMSsymbols$\rightrightarrows$ right right arrows; non-stretchy ⇉ class REL  \rightsquigarrow AMSsymbols$\rightsquigarrow$ right squiggle arrow; non-stretchy ⇝ class REL  \rightthreetimes AMSsymbols$\rightthreetimes$ right three times ⋌ class BIN  \risingdotseq AMSsymbols$\risingdotseq$ rising dot sequence ≓ class REL see also: \fallingdotseq \rlap  right overlap class ORD \rlap #1 creates a box of width zero; the argument is then placed just to the right of this zero-width box (and hence will overlap whatever lies to the right) Example:  a\mathrel{\rlap{\;/}{=}}b yields$a\mathrel{\rlap{\;/}{=}}b$In this example, {=} forces the equal to not have REL spacing (since it is not adjacent to ORD's); \mathrel{} forces the compound symbol (equal with overlapping slash) to be treated as a single REL; the \; improves the spacing for the slash. see also: \llap \rm  turns on roman; affects uppercase and lowercase letters, and digits; also affects uppercase Greek class ORD {\rm ... } Examples:  \rm AaBb\alpha\beta123 yields$\rm AaBb\alpha\beta123${\rm A B} A B yields${\rm A B} A B$\Delta\Gamma\Lambda{\rm\Delta\Gamma\Lambda} yields$\Delta\Gamma\Lambda{\rm\Delta\Gamma\Lambda}$\rm AB \bf CD yields$\rm AB \bf CD$\rm{AB}CD yields$\rm{AB}CD$see also: \text, \hbox, \mathrm \rmoustache$\rmoustache$ right moustache; non-stretchy when used alone; stretchy when used with \left or \right (see below) ⎱ class CLOSE Example:  \left\lmoustache \phantom{\matrix{a & b\cr c & d}} \right\rmoustache  yields $$\left\lmoustache \phantom{\matrix{a & b\cr c & d}} \right\rmoustache$$ see also: \lmoustache \root ... \of \root <index> \of #1 Examples:  \root 3 \of x yields$\root 3 \of x$\root 13 \of {\frac 12} yields$\root 13 \of {\frac 12}$\root n+1 \of x + 2 yields$\root n+1 \of x + 2 $see also: \sqrt, \leftroot, \uproot  \Rrightarrow AMSsymbols$\Rrightarrow$ non-stretchy ⇛ class REL  \Rsh AMSsymbols$\Rsh$ right shift; non-stretchy ↱ class REL see also: \Lsh  \rtimes AMSsymbols$\rtimes$ see also: \ltimes ⋊ class BIN \Rule (non-standard) a MathJax-specific macro giving a rule with a specified width, height, and depth \Rule <dimenWidth> <dimenHeight> <dimenDepth> where each argument is a dimension Examples:  x\Rule{3px}{1ex}{2ex}x yields$x\Rule{3px}{1ex}{2ex}x$x\Rule{3px}{2ex}{1ex}x yields$x\Rule{3px}{2ex}{1ex}x$ \rvert AMSmath \rVert AMSmath $\rvert\rVert$ ∣ class CLOSE ∥ class CLOSE both non-stretchy when used alone; stretchy when used with \left or \right Example:  \left\lvert\frac{\frac ab}{\frac cd}\right\rvert yields$\left\lvert\frac{\frac ab}{\frac cd}\right\rvert$see also: \lvert, \lVert, |, \| S \S$\S$ section symbol ꜀ class ORD \scr  turns on script typestyle for uppercase letters; lowercase letters are in a roman typestyle class ORD { \scr ... } Examples:  \scr ABCDEFGHIJKLMNOPQRSTUVWXYZ yields$\scr ABCDEFGHIJKLMNOPQRSTUVWXYZ$\scr 0123456789abcdefghijklmnopqrstuvwxyz yields$\scr 0123456789abcdefghijklmnopqrstuvwxyz$0123456789abcdefghijklmnopqrstuvwxyz yields$0123456789abcdefghijklmnopqrstuvwxyz${\scr AB}AB yields${\scr AB}AB$\scr AB \rm AB yields$\scr AB \rm AB$\scr{AB}CD yields$\scr{AB}CD$see also: \mathscr \scriptscriptstyle  used to over-ride automatic style rules and force scriptscript style; stays in force until the end of math mode or the braced group, or until another style is selected class ORD { \scriptscriptstyle ... } Example: In inline mode:  \frac ab+\displaystyle\frac ab+\textstyle\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab  yields:$\frac ab + \displaystyle\frac ab+\textstyle\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab$Example: In inline mode: \frac ab + {\scriptscriptstyle \frac cd + \frac ef} + \frac gh yields$\frac ab + {\scriptscriptstyle \frac cd + \frac ef} + \frac gh$Example: In inline mode: \frac ab + \scriptscriptstyle{\frac cd + \frac ef} + \frac gh yields$\frac ab + \scriptscriptstyle{\frac cd + \frac ef} + \frac gh$see also: \displaystyle, \scriptstyle, \textstyle \scriptsize   turns on script size class ORD { \scriptsize ... } Example:  \rm \scriptsize script \normalsize normal \large large yields$\rm \scriptsize script \normalsize normal \large large$see also: \normalsize \scriptstyle  used to over-ride automatic style rules and force script style; stays in force until the end of math mode or the braced group, or until another style is selected class ORD { \scriptstyle ... } Example: In inline mode:  \frac ab+\displaystyle\frac ab+\textstyle\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab  yields:$\frac ab + \displaystyle\frac ab+\textstyle\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab$Example: In inline mode: \frac ab + {\scriptstyle \frac cd + \frac ef} + \frac gh yields$\frac ab + {\scriptstyle \frac cd + \frac ef} + \frac gh$Example: In inline mode: \frac ab + \scriptstyle{\frac cd + \frac ef} + \frac gh yields$\frac ab + \scriptstyle{\frac cd + \frac ef} + \frac gh$see also: \displaystyle, \scriptscriptstyle, \textstyle \searrow$\searrow$ southeast arrow; non-stretchy ↘ class ORD see also: \nearrow, \nwarrow, \swarrow \sec$\sec$ secant; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples class OP Examples:  \sec x yields$\sec x$\sec(2x-1) yields$\sec(2x-1)$see also: \csc \setminus$\setminus$ set minus ∖ class BIN Examples:  A\setminus B yields$A\setminus B$A\backslash B yields$A\backslash B$see also: \backslash \sf  turns on sans serif mode for uppercase and lowercase letters and digits, and for uppercase Greek class ORD { \sf ... } Examples:  \sf ABCDEFGHIJKLMNOPQRSTUVWXYZ yields$\sf ABCDEFGHIJKLMNOPQRSTUVWXYZ$\sf 0123456789 yields$\sf 0123456789$\sf abcdefghijklmnopqrstuvwxyz yields$\sf abcdefghijklmnopqrstuvwxyz$ABCDE 01234 abcde yields$ABCDE 01234 abcde${\sf AB\Delta\Gamma\Lambda}\ AB\Delta\Gamma\Lambda yields${\sf AB\Delta\Gamma\Lambda}\ AB\Delta\Gamma\Lambda$\sf AB \rm AB yields$\sf AB \rm AB$\sf{AB}CD yields$\sf{AB}CD$see also: \mathsf \sharp$\sharp$ musical sharp symbol ♯ class ORD see also: \flat, \natural  \shortmid AMSsymbols$\shortmid$ see also: \nshortmid, \mid ∣ class REL  \shortparallel AMSsymbols$\shortparallel$ see also: \nshortparallel ∥ class REL  \shoveleft AMSmath \shoveright AMSmath forces flush left or flush right typesetting in a \multline or \multline* environment (see examples) Example: \begin{multline} (a+b+c+d)^2 \\ + (e+f)^2 + (g+h)^2 + (i+j)^2 + (k+l)^2 \\ + (m+n)^2 + (o+p)^2 + (q+r)^2 + (s+t)^2 + (u+v)^2 \\ + (w+x+y+z)^2 \end{multline}  yields $$\begin{multline} (a+b+c+d)^2 \\ + (e+f)^2 + (g+h)^2 + (i+j)^2 + (k+l)^2 \\ + (m+n)^2 + (o+p)^2 + (q+r)^2 + (s+t)^2 + (u+v)^2 \\ + (w+x+y+z)^2 \end{multline}$$ Example: \begin{multline} (a+b+c+d)^2 \\ \shoveleft{+ (e+f)^2 + (g+h)^2 + (i+j)^2 + (k+l)^2} \\ \shoveright{+ (m+n)^2 + (o+p)^2 + (q+r)^2 + (s+t)^2 + (u+v)^2} \\ + (w+x+y+z)^2 \end{multline}  yields $$\begin{multline} (a+b+c+d)^2 \\ \shoveleft{+ (e+f)^2 + (g+h)^2 + (i+j)^2 + (k+l)^2} \\ \shoveright{+ (m+n)^2 + (o+p)^2 + (q+r)^2 + (s+t)^2 + (u+v)^2} \\ + (w+x+y+z)^2 \end{multline}$$  \sideset AMSmath used for putting symbols at the four ‘corners’ of a large operator (like$\displaystyle\sum$or$\displaystyle\prod$) \sideset{_#1^#2}{_#3^#4} <large operator> where: • #1 = lower left • #2 = upper left • #3 = lower right • #4 = upper right Examples:  \sideset{_1^2}{_3^4}\sum yields $$\sideset{_1^2}{_3^4}\sum$$ $\sigma\Sigma$ lowercase Greek letter sigma σ class ORD uppercase Greek letter sigma Σ class ORD see also: \sum, \varsigma, \varSigma $\sim\simeq$ ∼ class REL ≃ class REL see also: \nsim \sin$\sin$ sine; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples class OP Examples:  \sin x yields$\sin x$\sin(2x-1) yields$\sin(2x-1)$see also: \cos \sinh$\sinh$ hyperbolic sine; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples class OP Examples:  \sinh x yields$\sinh x$\sinh(2x-1) yields$\sinh(2x-1)$see also: \cosh \skew used to finely adjust the positioning on accents; particularly useful for adjusting superaccents (accents on accents); usually requires trial-and-error adjustment for proper positioning \skew #1 <accent> where #1 is a positive integer (the skew amount) Examples:  \hat A yields$\hat A$\skew7\hat A yields$\skew7\hat A$\tilde M yields$\tilde M$\skew{8}\tilde M yields$\skew{8}\tilde M$\hat{\hat A} yields$\hat{\hat A}$\skew4\hat{\hat A} yields$\skew4\hat{\hat A}$\small  turns on small size; affects all math class ORD {\small ... } Example:  \rm\tiny tiny \Tiny Tiny \small small \normalsize normal \large lg \Large Lg \LARGE LG \huge hg \Huge Hg  yields$ \rm\tiny tiny \Tiny Tiny \small small \normalsize normal \large lg \Large Lg \LARGE LG \huge hg \Huge Hg $\def\myExp{\alpha\frac xy} \tiny\myExp \Tiny\myExp \small\myExp \normalsize\myExp \large\myExp \Large\myExp \LARGE\myExp \huge\myExp \Huge\myExp  yields$ \def\myExp{\alpha\frac xy} \tiny\myExp \Tiny\myExp \small\myExp \normalsize\myExp \large\myExp \Large\myExp \LARGE\myExp \huge\myExp \Huge\myExp $ab{\small cd} cd yields$ab{\small cd} cd$ab\small{cd} cd yields$ab\small{cd} cd$see also: \tiny, \Tiny, \normalsize, \large, \Large, \LARGE, \huge, \Huge  \smallfrown AMSsymbols$\smallfrown$ small frown ⌢ class REL see also: \frown, \smile, \smallsmile \smallint$\smallint$ small integral ∫ class OP see also: \int  \smallsetminus AMSsymbols$\smallsetminus$ small set minus ∖ class BIN see also: \setminus  \smallsmile AMSsymbols$\smallsmile$ small smile ⌣ class REL see also: \smile, \frown, \smallfrown \smash  By using \smash, \phantom, \hphantom, \vphantom, \rlap, \llap, you can typeset any mathematics, yet give it the width and/or height and/or depth of any other mathematics. \smash #1 Typesets the argument in a box with the same width as the argument, but with height and depth equal to zero. In other words: the argument of \smash is visible, and has its natural width, but does not contribute any height or depth to the surrounding mathematics (hence leaving the surrounding mathematics to dictate height and depth). class ORD Here are some scenarios: • to vertically \smash the box containing this and make it instead behave vertically like that : \smash{this}\vphantom{that} Examples:  \sqrt{\frac ab} \sqrt{\smash{7}\vphantom{\frac ab}}  yields$ \sqrt{\frac ab} \sqrt{\smash{7}\vphantom{\frac ab}} $\sqrt{\frac{\frac ab}{\frac cd}} \sqrt{\smash{\frac ef}\vphantom{\frac{\frac ab}{\frac cd}}}  yields$ \sqrt{\frac{\frac ab}{\frac cd}} \sqrt{\smash{\frac ef}\vphantom{\frac{\frac ab}{\frac cd}}} $• to horizontally compress the box containing this and make it instead behave horizontally like that : \rlap{this}\hphantom{that} or \hphantom{that}\llap{this} Examples:  \sqrt{\rm very\ wide} \sqrt{\rlap{\rm thin}\hphantom{\rm very\ wide}}  yields$ \sqrt{\rm very\ wide} \sqrt{\rlap{\rm thin}\hphantom{\rm very\ wide}} $\sqrt{\rm very\ wide} \sqrt{\hphantom{\rm very\ wide}\llap{\rm thin}}  yields$ \sqrt{\rm very\ wide} \sqrt{\hphantom{\rm very\ wide}\llap{\rm thin}} $• to both vertically smash and horizontally compress the box containing this and make it instead behave both vertically and horizontally like that : \rlap{\smash{this}}\phantom{that} or \phantom{that}\llap{\smash{this}} Examples:  \sqrt{\matrix{a & b\cr c & d}} \sqrt{ \rlap{\smash{\rm Hi!}} \phantom{\matrix{a & b\cr c & d}} }  yields$ \sqrt{\matrix{a & b\cr c & d}} \sqrt{ \rlap{\smash{\rm Hi!}} \phantom{\matrix{a & b\cr c & d}} } $see also: \hphantom, \vphantom, \phantom, \llap, \rlap \smile$\smile$ smile ⌣ class REL see also: \smallsmile, \frown, \smallfrown \space Example:  a\space b yields$a\space b$&#xA0; class ORD in MathJax, this is the same as: \ (backslash space), \nobreakspace \Space (non-standard) a MathJax-specific macro giving space with a specified width, height, and depth \Space <dimenWidth> <dimenHeight> <dimenDepth> where each argument is a dimension Compare:  a\Rule{5px}{4ex}{2ex}^b_c d yields$a\Rule{5px}{4ex}{2ex}^b_c d$a\Space{5px}{4ex}{2ex}^b_c d yields$a\Space{5px}{4ex}{2ex}^b_c d$see also: \Rule \spadesuit$\spadesuit$ see also: \clubsuit, \diamondsuit, \heartsuit ♠ class ORD  \sphericalangle AMSsymbols$\sphericalangle$ ∢ class ORD $\sqcap\sqcup$ square cap ⊓ class BIN square cup ⊔ class BIN \sqrt$\sqrt{}$ square root (and other roots) class ORD \sqrt #1 \sqrt[n]{op} is equivalent to \root n \of {op} Examples:  \sqrt x yields$\sqrt x$\sqrt xy yields$\sqrt xy$\sqrt{xy} yields$\sqrt{xy}$\sqrt[3]{x+1} yields$\sqrt[3]{x+1}$see also: \root  \sqsubset AMSsymbols \sqsupset AMSsymbols $\sqsubset\sqsupset$ square subset ⊏ class REL square superset ⊐ class REL $\sqsubseteq\sqsupseteq$ ⊑ class REL ⊒ class REL  \square AMSsymbols$\square$ □ class ORD \stackrel stack relations; you can stack anything (not just relations) but it creates an item of class REL (and usually the bottom is a REL to start with, but doesn't have to be) \stackrel #1 #2 where #1 (in superscript style) is stacked on top of #2 Examples:  \stackrel{\rm def}{=} yields$\stackrel{\rm def} {=}$\stackrel{\rm top}{\rm bottom} yields$\stackrel{\rm top}{\rm bottom}$\star$\star$ ⋆ class BIN \strut an invisible box with no width, height 8.6pt and depth 3pt; note that \mathstrut changes with the current size, but \strut does not Examples:  \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut}  yields$ \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut} $\Tiny \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut}  yields$ \Tiny \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut} $\Large \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut}  yields$ \Large \sqrt{(\ )} \sqrt{\mathstrut\rm mathstrut} \sqrt{\strut\rm strut} $see also: \mathstrut \style [HTML] non-standard; used to apply CSS styling to mathematics \style #1 #2 where: • #1 is a (single) CSS style declaration • #2 is the mathematics to be styled Examples:  \frac{\style{color:red}{x+1}}{y+2}  yields$\frac{\style{color:red}{x+1}}{y+2}$\style{background-color:yellow}{\frac{x+1}{y+2}}  yields$\style{background-color:yellow}{\frac{x+1}{y+2}}$Example: Consider the following HTML/Javascript/MathJax code: <button type="button" onclick="makeVisible()">Click to reveal answer</button> <script type="text/javascript"> function makeVisible() { document.getElementById('answer').style.visibility = "visible"; } </script> $$(x+1)^2 = \cssId{answer}\style{visibility:hidden}{(x+1)(x+1)}$$  Then, the result of this HTML/Javascript/MathJax code is: $$(x+1)^2 = \cssId{answer}{\style{visibility:hidden}{(x+1)(x+1)}}$$ see also: \class, \cssId \subset$\subset$ ⊂ class REL  \Subset AMSsymbols$\Subset$ ⋐ class REL  \subseteq \subsetneq AMSsymbols \subseteqq AMSsymbols \subsetneqq AMSsymbols $\subseteq\subsetneq\subseteqq\subsetneqq$ ⊆ class REL ⊊ class REL ⫅ class REL ⫋ class REL see also: \nsubseteq, \nsubseteqq, \varsubsetneq, \varsubsetneqq  \substack AMSmath use for multi-line subscripts or superscripts Examples:  \sum_{ \substack{ 1\lt i\lt 3 \\ 1\le j\lt 5 }} a_{ij}  yields (display mode) $$\sum_{ \substack{ 1\lt i\lt 3 \\ 1\le j\lt 5 }} a_{ij}$$ ^{\substack{\text{a very} \\ \text{contrived} \\ \text{example} }} {\frac ab}_{\substack{ \text{isn't} \\ \text{it?} }}  yields (display mode) $$^{\substack{\text{a very} \\ \text{contrived} \\ \text{example} }} {\frac ab}_{\substack{ \text{isn't} \\ \text{it?} }}$$ see also: \begin{subarray} \succ$\succ$ see also: \nsucc ≻ class REL  \succapprox AMSsymbols \succnapprox AMSsymbols $\succapprox\succnapprox$ ⪸ class REL ⪺ class REL  \succcurlyeq AMSsymbols$\succcurlyeq$ ≽ class REL  \succeq \succneqq AMSsymbols $\succeq\succneqq$ ⪰ class REL ⪶ class REL see also: \nsucceq  \succsim AMSsymbols \succnsim AMSsymbols $\succsim\succnsim$ ≿ class REL ⋩ class REL \sum$\sum$ summation notation; changes size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples ∑ class OP see also: \Sigma \sup$\sup$ supremum; greatest lower bound; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP Examples:  \sup_{\rm limit} yields (inline mode)$\sup_{\rm limit}$\sup_{\rm limit} yields (display mode)$\displaystyle\sup_{\rm limit}$see also: \inf \supset$\supset$ ⊃ class REL  \Supset AMSsymbols$\Supset$ ⋑ class REL  \supseteq \supsetneq AMSsymbols \supseteqq AMSsymbols \supsetneqq AMSsymbols $\supseteq\supsetneq\supseteqq\supsetneqq$ ⊇ class REL ⊋ class REL ⫆ class REL ⫌ class REL see also: \nsupseteq, \nsupseteqq, \varsupsetneq, \varsupsetneqq \surd$\surd$ √ class ORD \swarrow$\swarrow$ southwest arrow; non-stretchy ↙ class REL see also: \nearrow, \nwarrow, \searrow T  \tag AMSmath used primarily in AMS math environments to get tags (equation numbers, labels); can, however, be used on any equation; the argument of \tag is typeset in text mode, but math mode can be used within the text: for example, \tag{\$\bullet\} You can use dollar signs in text-mode regardless of the settings of the inlineMath delimiters in the tex2jax preprocessor. \tag #1 Example:  \eqalign{ 3x - 4y &= 5\cr x + 7 &= -2y } \tag{3.1c}  yields\eqalign{ 3x - 4y &= 5\cr x + 7 &= -2y } \tag{3.1c} $\tan$\tan$ tangent; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples class OP Examples:  \tan x yields$\tan x$\tan(2x-1) yields$\tan(2x-1)$see also: \cot \tanh$\tanh$ hyperbolic tangent; does not change size; default limit placement is the same in both inline and display modes; can change limit placement using \limits; see the Big Operators Table for more examples class OP Examples:  \tanh x yields$\tanh x$\tanh(2x-1) yields$\tanh(2x-1)$see also: \cosh, \sinh \tau$\tau$ lowercase Greek letter tau τ class ORD  \tbinom AMSmath notation commonly used for binomial coefficients; in textstyle \tbinom #1 #2 Examples:  \tbinom n k yields (inline mode)$\tbinom nk$\tbinom n k yields (display mode)$\displaystyle\tbinom nk$\binom n k yields (display mode)$\displaystyle\binom nk$\tbinom{n-1}k-1 yields$\binom{n-1}k-1$\tbinom{n-1}{k-1} yields$\tbinom{n-1}{k-1}$see also: \binom, \choose, \dbinom \TeX$\TeX$ the TeX logo class ORD Examples:  \TeX yields$\TeX$\rm\TeX yields$\rm\TeX$see also: \LaTeX  \text: text \textbf: boldface text \textit: italic text \textrm: roman text \textsf: sans serif text (added in MathJax 2.4) \texttt: typewriter text (added in MathJax 2.4) used to produce text-mode material (in a given font) within a mathematical expression; MathJax does not process any macros within the text (unlike$\rm\TeX$itself); you can get math mode within the text using $$...$$ delimiters class ORD \text #1 \textbf #1 \textit #1 \textrm #1 \textsf #1 \texttt #1 Examples:  |x| = x \text{ for all $$x \ge 0$$} yields$|x| = x \text{ for all $$x \ge 0$$}$\text{\alpha in text mode }\alpha yields$\text{\alpha in text mode }\alpha$\textbf{\alpha in textbf mode }\alpha yields$\textbf{\alpha in textbf mode }\alpha$\textit{\alpha in textit mode }\alpha yields$\textit{ \alpha in textit mode }\alpha$\textrm{\alpha in textrm mode }\alpha yields$\textrm{\alpha in textrm mode }\alpha$\textsf{\alpha in textsf mode }\alpha yields$\textsf{\alpha in textsf mode }\alpha$\texttt{\alpha in texttt mode }\alpha yields$\texttt{\alpha in texttt mode }\alpha$see also: \bf, \mathbf ; \it, \mathit ; \rm, \mathrm ; \sf, \mathsf ; \tt, \mathtt \textstyle  used to over-ride automatic style rules and force text (inline) style; stays in force until the end of math mode or the braced group, or until another style is selected class ORD { \textstyle ... } Example: In display mode: \frac ab + {\textstyle \frac cd + \frac ef} + \frac gh yields$\displaystyle\frac ab + {\textstyle \frac cd + \frac ef} + \frac gh$Example: In inline mode:  \frac ab+{\displaystyle\frac ab}+\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab  yields:$\frac ab + {\displaystyle\frac ab}+\frac ab+\scriptstyle\frac ab+\scriptscriptstyle\frac ab$see also: \displaystyle, \scriptstyle, \scriptscriptstyle  \tfrac AMSmath textstyle fraction \tfrac #1 #2 Examples:  \tfrac ab \frac ab (display mode) yields$\displaystyle\tfrac ab \frac ab$\tfrac ab \frac ab (inline mode) yields$\tfrac ab \frac ab$see also: \frac, \dfrac  \therefore AMSsymbols$\therefore$ ∴ class REL $\theta\Theta$ lowercase Greek letter theta θ class ORD uppercase Greek letter theta Θ class ORD see also: \vartheta, \varTheta  \thickapprox AMSsymbols$\thickapprox$ Example: approx\ \ \thickapprox yields$\approx\ \ \thickapprox $≈ class REL see also: \approx  \thicksim AMSsymbols$\thicksim$ Example: sim\ \ \thicksim yields$\sim\ \ \thicksim $∼ class REL \thinspace thin space; normally$\frac 16$of a quad Example: thinspaces between letters:$a\thinspace b\thinspace c\thinspace d$see also: symbols for spaces, \negthinspace \tilde$\tilde{}$ non-stretchy tilde accent ˜ \tilde #1 Usually, #1 is a single letter; otherwise, accent is centered over argument. Examples:  \tilde e yields$\tilde e$\tilde E yields$\tilde E$\tilde eu yields$\tilde eu$\tilde{eu} yields$\tilde{eu}$\times$\times$ × class BIN \tiny  turns on tiny; a bit smaller than \Tiny class ORD {\tiny ... } Examples:  \tiny AaBb\alpha\beta123 yields$\tiny AaBb\alpha\beta123${\tiny A B} A B yields${\tiny A B} A B$\tiny AB \Tiny CD yields$\tiny AB \Tiny AB$\tiny{AB}CD yields$\tiny{AB}CD$ \Tiny non-standard  turns on Tiny; a bit bigger than \tiny class ORD {\Tiny ... } Examples:  \Tiny AaBb\alpha\beta123 yields$\Tiny AaBb\alpha\beta123${\Tiny A B} A B yields${\Tiny A B} A B$\Tiny AB \tiny CD yields$\Tiny AB \tiny AB$\Tiny{AB}CD yields$\Tiny{AB}CD$\to$\to$ non-stretchy → class REL see also: \rightarrow tool tips Tool tips are not built into MathJax, but you can click here to benefit from a posting by Davide P. Cervone (April 2011) at the MathJax Users Group. \top$\top$ ⊤ class ORD  \triangle \triangledown AMSsymbols $\triangle\triangledown$ △ class ORD ▽ class ORD see also: \ntriangleleft, \ntriangleright, \vartriangle, \vartriangleleft, \vartriangleright $\triangleleft\triangleright$ ◃ class BIN ▹ class BIN see also: \ntriangleleft, \ntriangleright, \vartriangle, \vartriangleleft, \vartriangleright  \trianglelefteq AMSsymbols \trianglerighteq AMSsymbols $\trianglelefteq\trianglerighteq$ ⊴ class REL ⊵ class REL see also: \ntrianglelefteq, \ntrianglerighteq  \triangleq AMSsymbols$\triangleq$ ≜ class REL \tt  turns on typewriter type class ORD {\tt ... } Examples:  \tt AaBb\alpha\beta123 yields$\tt AaBb\alpha\beta123${\tt A B} A B yields${\tt A B} A B$\tt AB \rm CD yields$\tt AB \rm AB$\tt{AB}CD yields$\tt{AB}CD$ \twoheadleftarrow AMSsymbols \twoheadrightarrow AMSsymbols $\twoheadleftarrow\twoheadrightarrow$ non-stretchy ↞ class REL non-stretchy ↠ class REL U  \ulcorner AMSsymbols \urcorner AMSsymbols $\ulcorner\urcorner$ upper left corner ┌ class REL upper right corner ┐ class REL These are technically delimiters, but MathJax doesn't stretch them. They are valid after \left, \right, and the various \big commands. see also: \llcorner, \lrcorner \underbrace puts a (stretchy) under-brace under the argument; can use ‘^’ to place an optional superscript over the argument; can use ‘_’ to place an optional subscript below the underbrace \underbrace #1 Example:  \underbrace{x + \cdots + x}_{n\rm\ times}^{\text{(note here)} yields$\underbrace{x + \cdots + x}_{n\rm\ times}^{\text{(note here)}}$see also: \overbrace $\underleftarrow{}\underrightarrow{}\underleftrightarrow{}$ stretchy under left arrow ← stretchy under right arrow → stretchy under left right arrow ↔ \underleftarrow #1 \underrightarrow #1 \underleftrightarrow #1 Examples:  \underleftarrow{\text{the argument}} yields$\underleftarrow{\text{the argument}}$\underrightarrow{AB} yields$\underrightarrow{AB}$\underrightarrow{AB\strut} yields$\underrightarrow{AB\strut}$\underleftrightarrow{\hspace1in} yields$\underleftrightarrow{\hspace1in}$\underline$\underline{}$ stretchy underline _ \underline #1 Examples:  \underline{AB} yields$\underline{AB}$\underline a yields$\underline a$\underline{\text{a long argument}} yields$\underline{\text{a long argument}}$\underparen puts a (stretchy) under-parenthesis (under-arc, smile) under the argument (new in MathJax 2.6) \underparen #1 Example:  \underparen a \quad \underparen ab \quad \underparen{ab} \quad \underparen{abc} \quad \underparen{abcdef} \quad \underparen{\overparen{abcd}}  yields$\underparen a \quad \underparen ab \quad \underparen{ab} \quad \underparen{abc} \quad \underparen{abcdef} \quad \underparen{\overparen{abcd}} see also: \overparen, \smallfrown, \frown, \smallsmile, \smile \underset \underset #1 #2 undersets argument #1 (in scriptstyle) under argument #2; the top item is properly aligned with the surrounding text (their baselines match) Examples:  \underset{\rm bottom}{\rm top} yields\underset{\rm bottom}{\rm top}$\underset ab yields $$\underset ab$$ see also: \overset  \unicode non-standard  implements a \unicode{} extension to$\rm\TeX$that allows arbitrary unicode code points to be entered in mathematics; can optionally specify height and depth of character (width is determined by browser); can optionally specify the default font from which to take the character; once a size and font are provided for a given unicode point, they need not be specified again in subsequent \unicode{} calls for that character class ORD \unicode[optHeight,optDepth][optFont]#1 Examples:  \unicode{x263a} yields$\unicode{x263a}$☺ yields (in math mode)$☺$\unicode[.55,0.05]{x22D6} yields$\unicode[.55,0.05]{x22D6}$less-than with dot, withheight 0.55em and depth 0.05em \unicode[.55,0.05][Geramond]{x22D6} yields$\unicode[.55,0.05][Geramond]{x22D6}$same, taken from Geramond font \unicode[Geramond]{x22D6} yields$\unicode[Geramond]{x22D6}$same, but with default (height,depth) of (0.8em,0.2em)  \unlhd AMSsymbols \unrhd AMSsymbols $\unlhd\unrhd$ underlined left-hand (left-pointing) diamond ⊴ class REL underlined right-hand (right-pointing) diamond ⊵ class REL $\uparrow\Uparrow$ non-stretchy ↑ class REL non-stretchy ⇑ class REL $\updownarrow\Updownarrow$ non-stretchy ↕ class REL non-stretchy ⇕ class REL  \upharpoonleft AMSsymbols \upharpoonright AMSsymbols $\upharpoonleft\upharpoonright$ non-stretchy ↿ class REL non-stretchy ↾ class REL \uplus$\uplus$ ⊎ class BIN \uproot used to fine-tune the placement of the index inside \sqrt or \root (see examples) \sqrt[... \uproot #1 ...]{...} \root ... \uproot #1 ... \of {...} where the argument is a small integer: a positive integer moves the index up; a negative integer moves the index down In actual TeX, \uproot is not allowed in \root , so this is a difference between MathJax and$\rm\TeX$. Examples:  \sqrt[3]{x} yields$\sqrt[3]{x}$\sqrt[3\uproot2]{x} yields$\sqrt[3\uproot2]{x}$\root 3 \of x yields$\root 3 \of x$\root 3\uproot{-2} \of x yields$\root 3\uproot{-2} \of x$see also: \leftroot, \root $\upsilon\Upsilon$ lowercase Greek letter upsilon υ class ORD uppercase Greek letter upsilon Υ class ORD see also: \varupsilon, \varUpsilon  \upuparrows AMSsymbols$\upuparrows$ non-stretchy ⇈ class REL V  \varDelta AMSsymbols$\varDelta$ uppercase Greek letter delta; variant Δ class ORD see also: \Delta  \varepsilon$\varepsilon$ lowercase Greek letter epsilon; variant ε class ORD see also: \epsilon  \varGamma AMSsymbols$\varGamma$ uppercase Greek letter gamma; variant Γ class ORD see also: \Gamma  \varinjlim AMSmath$\varinjlim$ injective limit; variant; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples class OP see also: \injlim  \varkappa AMSsymbols$\varkappa$ lowercase Greek letter kappa; variant ϰ class ORD see also: \kappa  \varLambda AMSsymbols$\varLambda$ uppercase Greek letter lambda; variant Λ class ORD see also: \Lambda  \varlimsup AMSmath \varliminf AMSmath $\varlimsup\varliminf$ limit superior; variant class OP limit inferior; variant class OP do not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples see also: \limsup, \liminf  \varnothing AMSsymbols$\varnothing$ see also: \emptyset ∅ class ORD  \varOmega AMSsymbols$\varOmega$ uppercase Greek letter omega; variant Ω class ORD see also: \Omega \varphi$\varphi$ lowercase Greek letter phi; variant φ class ORD see also: \phi  \varPhi AMSsymbols$\varPhi$ uppercase Greek letter phi; variant Φ class ORD see also: \Phi \varpi$\varpi$ lowercase Greek letter pi; variant ϖ class ORD see also: \pi  \varPi AMSsymbols$\varPi$ uppercase Greek letter pi; variant Π class ORD see also: \Pi  \varprojlim AMSmath$\varprojlim$projective limit; variant; does not change size; can change limit placement using \limits and \nolimits; see the Big Operators Table for examples see also: \projlim  \varpropto AMSsymbols$\varpropto$ proportional to; variant ∝ class REL see also: \propto  \varPsi AMSsymbols$\varPsi$ uppercase Greek letter pi; variant Ψ class ORD see also: \Psi  \varrho AMSsymbols$\varrho$ lowercase Greek letter rho; variant ϱ class ORD see also: \rho  \varsigma AMSsymbols$\varsigma$ lowercase Greek letter sigma; variant ς class ORD see also: \sigma  \varSigma AMSsymbols$\varSigma$ uppercase Greek letter sigma; variant ς class ORD see also: \Sigma  \varsubsetneq AMSsymbols \varsubsetneqq AMSsymbols $\varsubsetneq\varsubsetneqq$ ⊊ class REL ⫋ class REL see also: \subsetneq, \subsetneqq  \varsupsetneq AMSsymbols \varsupsetneqq AMSsymbols $\varsupsetneq\varsupsetneqq$ ⊋ class REL ⫌ class REL see also: \supsetneq, \supsetneqq  \vartheta \varTheta AMSsymbols $\vartheta\varTheta$ lowercase Greek letter theta; variant ϑ class ORD uppercase Greek letter theta; variant Θ class ORD see also: \theta, \Theta  \vartriangle AMSsymbols \vartriangleleft AMSsymbols \vartriangleright AMSsymbols $\vartriangle\vartriangleleft\vartriangleright$ △ class REL ⊲ class REL ⊳ class REL see also: \triangle, \triangleleft, \triangleright  \varUpsilon AMSsymbols$\varUpsilon$ uppercase Greek letter upsilon; variant Υ class ORD see also: \upsilon  \varXi AMSsymbols$\varXi$ uppercase Greek letter xi; variant Ξ class ORD see also: \Xi \vcenter \vcenter #1 centers the argument on the ‘math axis’, which is at half the height of an ‘x’, or about the position of a minus sign; one of the reasons for \vcenter is to get stretchy delimiters to match the contents better Examples:  \left(\Rule{1ex}{2em}{0pt}\right) yields$\left(\Rule{1ex}{2em}{0pt}\right)$\left(\vcenter{\Rule{1ex}{2em}{0pt}}\right) yields$\left(\vcenter{\Rule{1ex}{2em}{0pt}}\right)$\left(\frac{a+b}{\dfrac{c}{d}}\right) yields $$\left(\frac{a+b}{\dfrac{c}{d}}\right)$$ \left(\vcenter{\frac{a+b}{\dfrac{c}{d}}}\right) yields $$\left(\vcenter{\frac{a+b}{\dfrac{c}{d}}}\right)$$ \vdash$\vdash$ see also: \nvdash ⊢ class REL  \Vdash AMSsymbols \vDash AMSsymbols $\Vdash\vDash$ ⊩ class REL ⊨ class REL see also: \nVdash, \nvDash \vdots$\vdots$ vertical dots ⋮ class ORD \vec non-stretchy vector symbol \vec #1 Examples:  \vec v yields$\vec v$\vec{AB} yields$\vec{AB}$see also: \overrightarrow \vee$\vee$ see also: \lor ∨ class BIN  \veebar AMSsymbols$\veebar$ ⊻ class BIN \verb verbatim mode; useful for code snippets and for displaying special characters ‘as is’ (i.e., not interpreted by MathJax). Only works in display mode. Usually, verbatim content is typeset in a sans serif font. \verb$\diamond$<non-interpreted material>$\diamond$where$\diamond$denotes a non-letter character that does not appear in the <non-interpreted material>. To use \verb : • First look through the material that is to be typeset ‘as is’ (verbatim). • Choose a non-letter character that does not appear in this material. • This chosen non-letter character will mark the beginning and end of the verbatim material, as illustrated in the examples below. Examples (in display mode): \verb*$x^2\sqrt y$* \text{ yields } x^2\sqrt y yields: $$\verb*x^2\sqrt y* \text{ yields } x^2\sqrt y$$ \verb!Text and$\frac ab$in \verb mode! yields: $$\verb!Text and \frac ab in \verb mode!$$ $\vert\Vert$ class ORD ∥ class ORD both non-stretchy when used alone; stretchy when used with \left or \right see also: |, \|, \lvert, \lVert, \rvert, \rVert \vphantom vertical phantom Sometimes you want to pretend that something is there, for spacing reasons, but you don't want it to appear—you want it to be invisible—you want it to be a phantom. The box created by \vphantom has the height and depth of its argument, but its width is zero (so it doesn't contribute to any horizontal spacing issues). In other words, \vphantom creates vertical space equal to that produced by its argument, but doesn't create any horizontal space. \vphantom #1 Examples:  \binom{\frac ab}c \binom{\vphantom{\frac ab}?}c yields $$\binom{\frac ab}c \binom{\vphantom{\frac ab}?}c$$ see also: \phantom, \hphantom, \smash  \Vvdash AMSsymbols$\Vvdash$ ⊪ class REL W \wedge$\wedge$ see also: \land ∧ class BIN \widehat$\widehat{\ \ \ }$ stretchy hat accent ˆ \widehat #1 Examples:  \widehat a yields$\widehat a$\widehat A yields$\widehat A$\widehat AB yields$\widehat AB$\widehat{AB} yields$\widehat{AB}$see also: \hat \widetilde$\widetilde{\ \ \ }$ stretchy tilde accent ˜ \widetilde #1 Examples:  \widetilde a yields$\widetilde a$\widetilde A yields$\widetilde A$\widetilde AB yields$\widetilde AB$\widetilde{AB} yields$\widetilde{AB}$\wp$\wp$ ‘wriggly’ letter p ℘ class ORD \wr$\wr$ ‘wriggle’ symbol; ≀ class BIN X \Xi$\Xi$ uppercase Greek letter xi Ξ class ORD see also: \varXi \xi$\xi$ lowercase Greek letter xi ξ class ORD  \xleftarrow AMSmath \xrightarrow AMSmath  stretchy arrows with mathematical overset and optional mathematical underset class REL \xleftarrow[optionalArgument] #1 \xrightarrow[optionalArgument] #1 where the optional arguments (inside brackets, if desired) appear below the arrows (see examples). Examples:  \xrightarrow a yields$\xrightarrow a$\xrightarrow ab yields$\xrightarrow ab$\xrightarrow{ab} yields$\xrightarrow{ab}$\xleftarrow{\text{see equation (1)}} yields$\xleftarrow{\text{see equation (1)}}$\xrightarrow[f]{\text{see (1)}} yields$\xrightarrow[f]{\text{see (1)}}$Y  \yen AMSsymbols$\yen$ ¥ class ORD Z \zeta$\zeta$ lowercase Greek letter zeta ζ class ORD environments$\rm\LaTeXenvironments of the form \begin{XXX} ... \end{XXX} are provided, as listed in the table below. The processEnvironments value in the tex2jax block of the MathJax configuration controls processing behavior: • processEnvironments: true (the default) causes environments to be processed both inside and outside of math delimiters • processEnvironments: false causes environments to be processed only when they appear inside math delimiters  align AMSmath \begin{align} ... \end{align}  For vertical alignment of two or more lines at one or more places: • ampersand(s) ‘&’ are used to indicate desired alignments (see examples below) • a double backslash ‘\\’ or carriage return ‘\cr’ separates lines • individual lines may be tagged using the \tag{} command: • default input for \tag{} is text • you may get mathematical content inside \tag{} by using math delimiters; e.g., \tag{\alpha} EXAMPLES: Alignment at a single location: • use a single ampersand where alignment should occur • you may tag (or not tag) any desired subset of lines \begin{align} (a+b)^2 &= (a+b)(a+b) \tag{3.1c} \\ &= a^2 + ab + ba + b^2 \tag{\dagger$} \\ &= a^2 + 2ab + b^2 \tag{$\ast} \end{align}  yields \begin{align} (a+b)^2 &= (a+b)(a+b) \tag{3.1c} \\ &= a^2 + ab + ba + b^2 \tag{\dagger} \\ &= a^2 + 2ab + b^2 \tag{\ast} \end{align} Alignment at more than one location is trickier. It is best illustrated with an example: see also: \eqalign, \eqalignno, \leqalignno  align* AMSmath [May 2011] same as align  alignat AMSmath \begin{alignat}{} ... \end{alignat} For vertical alignment of two or more lines at one or more places; produces a more horizontally-compressed display than align: • the alignat environment is started with \begin{alignat}{<num>} , where num is a positive integer (1,2,3,\ldots) that indicates the number of places where alignment is desired • ampersand(s) ‘&’ are used to indicate desired alignments (see examples below) • a double backslash ‘\\’ or carriage return ‘\cr’ separates lines • individual lines may be tagged using the \tag{} command: • default input for \tag{} is text • you may get mathematical content inside \tag{} by using math delimiters; e.g., \tag{\alpha$} Let$ndenote the number of places where alignment is desired. Then, there will be2n - 1ampersands used, as follows: • STEP 1: The odd-numbered ampersands (1st, 3rd, 5th, etc.) are placed where alignment is desired. Position these ampersands first: a &= bbbbbb &= cc &= d \\ aaa &= bbbb &= cccccc &= ddd  • STEP 2: Now, focus attention on the content between the previously-positioned ampersands. What part of this content belongs on the left? On the right? In each group, use an ampersand to separate the content into two pieces (a piece may be empty). Think of this ampersand as a solid ‘wall’ that is pushing content to the left or right. Compare these three scenarios: Pushing all content to the left: \begin{alignat}{3} a &= bbbbbb& &= cc& &= d \tag{3.1} \\ aaa &= bbbb& &= cccccc& &= ddd \tag{3.2} \end{alignat}  yields \begin{alignat}{3} a &= bbbbbb& &= cc& &= d \tag{3.1}\\ aaa &= bbbb& &= cccccc& &= ddd \tag{3.2} \end{alignat} Pushing all content to the right: \begin{alignat}{3} a &= &bbbbbb &= &cc &= d \\ aaa &= &bbbb &= &cccccc &= ddd \end{alignat}  yields \begin{alignat}{3} a &= &bbbbbb &= &cc &= d \\ aaa &= &bbbb &= &cccccc &= ddd \end{alignat} Splitting the content, with half left and half right: \begin{alignat}{3} a &= bbb&bbb &= c&c &= d \\ aaa &= bb&bb &= ccc&ccc &= ddd \end{alignat}  yields \begin{alignat}{3} a &= bbb&bbb &= c&c &= d \\ aaa &= bb&bb &= ccc&ccc &= ddd \end{alignat} see also: \eqalignat, \eqalignatno, \leqalignatno  alignat* AMSmath [May 2011] same as alignat  array \begin{array}{} ... \end{array}  Used to create an array (matrix), where columns can be individually left-justified, centered, or right-justified. • suppose thatn$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • the array environment is started with \begin{array}{<justification info>} , where <justification info> is a series of$n$letters, one for each column: • ‘l’ for left-justified • ‘c’ for centered • ‘r’ for right-justified • pipe character(s) ‘|’ can be used in the justification information to specify optional separating vertical line(s) (see example below) • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Compare these scenarios: both columns left-justified: \begin{array}{ll} aaa & b\cr c & ddd \end{array}  yields $$\begin{array}{ll} aaa & b\cr c & ddd \end{array}$$ both columns right-justified: \begin{array}{rr} aaa & b\cr c & ddd \end{array}  yields $$\begin{array}{rr} aaa & b\cr c & ddd \end{array}$$ both columns centered, with separating line: \begin{array}{c|c} aaa & b\cr c & ddd \end{array}  yields $$\begin{array}{c|c} aaa & b\cr c & ddd \end{array}$$ first column left-justified; second column right-justified: \begin{array}{lr} aaa & b\cr c & ddd \end{array}  yields $$\begin{array}{lr} aaa & b\cr c & ddd \end{array}$$ Putting a pipe character ‘|’ at the beginning or end of the justification info encloses the entire structure, which is different from standard$\,\rm\TeX\,$: \begin{array}{|lr} aaa & b\cr c & ddd \end{array}  yields $$\begin{array}{|lr} aaa & b\cr c & ddd \end{array}$$ see also: \begin{matrix}, \begin{subarray}  Bmatrix \begin{Bmatrix} ... \end{Bmatrix} Used to create a matrix (an array) with braces$\{\,,\,\}$as enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  \begin{Bmatrix} aaa & b\cr c & ddd \end{Bmatrix}  yields $$\begin{Bmatrix} aaa & b\cr c & ddd \end{Bmatrix}$$ see also: \begin{array}, \begin{matrix}  bmatrix \begin{bmatrix} ... \end{bmatrix} Used to create a matrix (an array) with brackets$[\,,\,]$as enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  \begin{bmatrix} aaa & b\cr c & ddd \end{bmatrix}  yields $$\begin{bmatrix} aaa & b\cr c & ddd \end{bmatrix}$$ see also: \begin{array}, \begin{matrix}  cases \begin{cases} ... \end{cases} Used for piecewise-defined functions • an ampersand ‘&’ is used to separate the function cases and their definitions • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  |x| = \begin{cases} x & \text{ if } x\ge 0 \\ -x & \text{ if } x \lt 0 \end{cases}  yields $$|x| = \begin{cases} x & \text{ if } x\ge 0 \\ -x & \text{ if } x \lt 0 \end{cases}$$ see also: \cases  eqnarray \begin{eqnarray} ... \end{eqnarray} for ‘equation arrays’; aligns at one or more places; surround the character(s) to be aligned with ampersands, as shown below; content between alignment characters (or between alignment characters and end-of-line) is left-justified; a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Examples: \begin{eqnarray} y &=& (x-1)^2 \\ &=& (x-1)(x-1) \\ &=& x^2 - 2x + 1 \end{eqnarray}  yields $$\begin{eqnarray} y &=& (x-1)^2 \\ &=& (x-1)(x-1) \\ &=& x^2 - 2x + 1 \end{eqnarray}$$ \begin{eqnarray} (x-1)^2 &=& (x-1)(x-1) &=& x^2-2x + 1 \\ (x-1)^3 &=& (x-1)(x-1)(x-1) &=& (x-1)^2(x-1) \end{eqnarray}  yields $$\begin{eqnarray} (x-1)^2 &=& (x-1)(x-1) &=& x^2-2x + 1 \\ (x-1)^3 &=& (x-1)(x-1)(x-1) &=& (x-1)^2(x-1) \end{eqnarray}$$ eqnarray* [May 2011] same as equarray  equation ... [May 2011] ignored, until MathJax implements automatic numbering equation* [May 2011] ignored  gather AMSmath to display any number of centered formulas (without any alignment); a double backslash ‘\\’ or carriage return ‘\cr’ separates rows; individual lines may be tagged using the \tag{} command: • default input for \tag{} is text • you may get mathematical content inside \tag{} by using math delimiters; e.g., \tag{\alpha$} Example: \begin{gather} a = a \tag{$*$}\\ \text{if } a=b \text{ then } b=a \tag{$\dagger$}\\ \text{if } a=b \text{ and } b=c \text{ then } a=c\tag{3.1} \end{gather}  yields: $$\begin{gather} a = a \tag{*}\\ \text{if } a=b \text{ then } b=a \tag{\dagger}\\ \text{if } a=b \text{ and } b=c \text{ then } a=c \tag{3.1} \end{gather}$$ see also: \displaylines  gather* AMSmath [May 2011] same as gather  matrix \begin{matrix} ... \end{matrix} Used to create a matrix (an array) without any enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  \begin{matrix} aaa & b\cr c & ddd \end{matrix}  yields $$\begin{matrix} aaa & b\cr c & ddd \end{matrix}$$ see also: \begin{array}  multline AMSmath \begin{multline} ... \end{multline} a multi-line environment; typically used for formulas/equations that don't fit on a single line • the first (or only) line is displayed left-justified • the last line is displayed right-justified • any intermediate line(s) are centered The justification of intermediate lines can be adjusted with \shoveleft and \shoveright. Examples: \begin{multline} \rm first\ line \\ \rm second\ line \\ \rm third\ line \\ \rm fourth\ line \end{multline}  yields: $$\begin{multline} \rm first\ line \\ \rm second\ line \\ \rm third\ line \\ \rm fourth\ line \end{multline}$$ \begin{multline} \rm first\ line \\ \shoveleft\rm second\ line \\ \shoveright\rm third\ line \\ \rm fourth\ line \end{multline}  yields: $$\begin{multline} \rm first\ line \\ \shoveleft\rm second\ line \\ \shoveright\rm third\ line \\ \rm fourth\ line \end{multline}$$ see also: \begin{split} multline* [AMSmath] [May 2011] same as multline see also: \shoveleft, \shoveright  pmatrix \begin{pmatrix} ... \end{pmatrix} Used to create a matrix (an array) with parentheses$(\,,\,)$as enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  \begin{pmatrix} aaa & b\cr c & ddd \end{pmatrix}  yields $$\begin{pmatrix} aaa & b\cr c & ddd \end{pmatrix}$$ see also: \begin{array}, \begin{matrix}  smallmatrix AMSmath \begin{smallmatrix} ... \end{smallmatrix} Used to create a small matrix (an array); particularly suited for use in text; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Examples:  the matrix$\begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix}$is...  yields the matrix$ \begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix} $is... \left[ \begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix} \right]  yields (in display mode) $$\left[ \begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix} \right]$$ \left[ \begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix} \right]  yields (in inline mode)$ \left[ \begin{smallmatrix} aaa & b\cr c & ddd \end{smallmatrix} \right] see also: \begin{array}, \begin{matrix}  split AMSmath for single equations that are too long to ﬁt on one line, and hence must be split into multiple lines; allows for (optional) alignment at one or more places, using ‘&’ to mark alignment points Examples: \begin{split} \text{first line}\\ &\text{first aligned place} &\text{second aligned place} \\ &\text{and more first aligned}\qquad &\text{and more second aligned} \\ \text{no ampersands on this line} \\ & &\text{aligned at second place} \\ \text{no amps here either} \end{split}  yields: \begin{split} \text{first line}\\ &\text{first aligned place} &\text{second aligned place} \\ &\text{and more first aligned}\qquad &\text{and more second aligned} \\ \text{no ampersands on this line} \\ & &\text{aligned at second place} \\ \text{no amps here either} \end{split} see also: \begin{multline}  subarray \begin{subarray}{} ... \end{subarray}  a more compact version of \begin{array}; can be used for multi-subscripts and multi-superscripts on large operators; columns can be individually left-justified, centered, or right-justified • suppose thatn$columns are desired in the subarray; then,$n-1$ampersands are used to separate the columns • the subarray environment is started with \begin{subarray}{<justification info>} , where <justification info> is a series of$n$letters, one for each column: • ‘l’ for left-justified • ‘c’ for centered • ‘r’ for right-justified • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example: \prod_{\begin{subarray}{rl} i\lt 5 & j\gt 1 \\ k\ge2,\,k\ne 5 \quad & \ell\le 5,\,\ell\ne 2 \end{subarray}} x_{ijk\ell}  yields $$\prod_{\begin{subarray}{rl} i\lt 5\quad & j\gt 1 \\ k\ge2,\,k\ne 5 \quad & \ell\le 5,\,\ell\ne 2 \end{subarray}} x_{ijk\ell}$$ see also: \substack, \begin{array}  Vmatrix \begin{Vmatrix} ... \end{Vmatrix} Used to create a matrix (an array) with$\|\,,\,\|$as enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1$ampersands are used to separate the columns • a double backslash ‘\\’ or carriage return ‘\cr’ separates rows Example:  \begin{Vmatrix} aaa & b\cr c & ddd \end{Vmatrix}  yields $$\begin{Vmatrix} aaa & b\cr c & ddd \end{Vmatrix}$$ see also: \begin{array}, \begin{matrix}  vmatrix \begin{vmatrix} ... \end{vmatrix} Used to create a matrix (an array) with$|\,,\,|$as enclosing delimiters; columns are centered. • suppose that$n$columns are desired in the array; then,$n-1\$ ampersands are used to separate the columns
• a double backslash ‘\\’ or carriage return ‘\cr’ separates rows
Example:
 \begin{vmatrix} aaa & b\cr c & ddd \end{vmatrix}  yields $$\begin{vmatrix} aaa & b\cr c & ddd \end{vmatrix}$$