.. include:: ../../header2.txt ============================ LaTeX syntax for mathematics ============================ .. role:: m(math) .. default-role:: math .. |latex| replace:: L\ :sup:`A`\ T\ :sub:`E`\ X :abstract: Docutils supports mathematical content with a `"math" directive`__ and `role`__. The input format is *LaTeX math syntax*\ [#math-syntax]_ with support for literal Unicode symbols. .. sectnum:: .. contents:: __ https://docutils.sourceforge.io/docs/ref/rst/directives.html#math __ https://docutils.sourceforge.io/docs/ref/rst/roles.html#math Inline formulas and displayed equations ======================================= The **math role** can be used for inline mathematical expressions: ``:math:`\psi(r) = \exp(-2r)``` will produce :m:`\psi(r)=\exp(-2r)`. Inside the backtics you can write anything you would write between dollar signs in a LaTeX document. [#math-syntax]_   .. tip:: If you put ``.. default-role:: math`` at the top of your document, you can write ```x^2``` instead of the longer version: ``:math:`x^2```. You can also introduce an abbreviation like this ``.. role:: m(math)``. That will allow you to write ``:m:`x^2``` or ```x^2`:m:``. The **math directive** is used for displayed equations. It corresponds to an ``equation*`` or ``align*`` environment in a LaTeX document. If you write:: .. math:: \psi(r) = e^{-2r} you will get: .. math:: \psi(r) = e^{-2r} A more complex example is the definition of the `Fourier transform`_:: .. math:: :name: Fourier transform (\mathcal{F}f)(y) = \frac{1}{\sqrt{2\pi}^{\ n}} \int_{\mathbb{R}^n} f(x)\, e^{-\mathrm{i} y \cdot x} \,\mathrm{d} x. which is rendered as: .. math:: :name: Fourier transform (\mathcal{F}f)(y) = \frac{1}{\sqrt{2\pi}^{\ n}} \int_{\mathbb{R}^n} f(x)\, e^{-\mathrm{i} y \cdot x} \,\mathrm{d} x. The ``:name:`` option puts a label on the equation that can be linked to by `hyperlink references`_. Displayed equations can use ``\\`` and ``&`` for line shifts and alignments:: .. math:: a &= (x + y)^2 & b &= (x - y)^2 \\ &= x^2 + 2xy + y^2 & &= x^2 - 2xy + y^2 LaTeX output will wrap it in an ``align*`` environment. The result is: .. math:: a & = (x + y)^2 & b & = (x - y)^2 \\ & = x^2 + 2xy + y^2 & & = x^2 - 2xy + y^2 The ``aligned`` environment can be used as a component in a containing expression. E.g., :: .. math:: \left. \begin{aligned} B' & = -\partial\times E, \\ E' & = \partial\times B - 4\pi j, \end{aligned} \right\} \qquad \text{Maxwell’s equations} results in .. math:: \left. \begin{aligned} B' & = -\partial\times E \\ E' & = \partial\times B - 4\pi j \end{aligned} \;\right\} \qquad \text{Maxwell’s equations.} .. [#math-syntax] The supported LaTeX commands include AMS extensions (see, e.g., the `Short Math Guide`_). Some of the shown symbols require the "amssymb" `LaTeX package`_ (or another package providing the AMS symbol macros) when exported with the "latex" writer. The support is limited to a subset of *LaTeX math* by the conversion required for many output formats. For HTML, the `math_output`_ configuration setting (or the corresponding ``--math-output`` command line option) selects between alternative output formats with different subsets of supported elements. If a writer does not support math typesetting, the content is inserted verbatim. .. _hyperlink references: ../ref/rst/restructuredtext.html#hyperlink-references .. _Short Math Guide: https://mirrors.ctan.org/info/short-math-guide/short-math-guide.pdf .. _math_output: https://docutils.sourceforge.io/docs/user/config.html#math-output .. _LaTeX package: ../../user/latex.html#latex-document-classes-and-packages Mathematical symbols ==================== The following tables are adapted from the first edition of "The LaTeX Companion" (Goossens, Mittelbach, Samarin) and the AMS `Short Math Guide`_. Accents and embellishments -------------------------- The "narrow" accents are intended for a single-letter base. .. class:: colwidths-auto =========== ============= ============ ============== ============== ================ ========= =========== `\acute{x}` ``\acute{x}`` `\dot{t}` ``\dot{t}`` `\grave{x}` ``\grave{x}`` `\vec{x}` ``\vec{x}`` `\bar{v}` ``\bar{v}`` `\ddot{t}` ``\ddot{t}`` `\hat{x}` ``\hat{x}`` `\breve{x}` ``\breve{x}`` `\dddot{t}` ``\dddot{t}`` `\mathring{x}` ``\mathring{x}`` `\check{x}` ``\check{x}`` `\ddddot{t}` ``\ddddot{t}`` `\tilde{n}` ``\tilde{n}`` =========== ============= ============ ============== ============== ================ ========= =========== When adding an accent to an i or j in math, dotless variants can be obtained with ``\imath`` and ``\jmath``: `\hat \imath`, `\vec{\jmath}`. For embellishments that span multiple symbols, use: .. class:: colwidths-auto ========================== ============================ =========================== ============================= `\widetilde{gbi}` ``\widetilde{gbi}`` `\widehat{gbi}` ``\widehat{gbi}`` `\overline{gbi}` ``\overline{gbi}`` `\underline{gbi}` ``\underline{gbi}`` `\overbrace{gbi}` ``\overbrace{gbi}`` `\underbrace{gbi}` ``\underbrace{gbi}`` `\overleftarrow{gbi}` ``\overleftarrow{gbi}`` `\underleftarrow{gbi}` ``\underleftarrow{gbi}`` `\overrightarrow{gbi}` ``\overrightarrow{gbi}`` `\underrightarrow{gbi}` ``\underrightarrow{gbi}`` `\overleftrightarrow{gbi}` ``\overleftrightarrow{gbi}`` `\underleftrightarrow{gbi}` ``\underleftrightarrow{gbi}`` ========================== ============================ =========================== ============================= Binary operators ---------------- .. class:: colwidths-auto ================== ==================== ================= =================== ================== ==================== `*` ``*`` `\circledast` ``\circledast`` `\ominus` ``\ominus`` `+` ``+`` `\circledcirc` ``\circledcirc`` `\oplus` ``\oplus`` `-` ``-`` `\circleddash` ``\circleddash`` `\oslash` ``\oslash`` `:` ``:`` `\cup` ``\cup`` `\otimes` ``\otimes`` `\Cap` ``\Cap`` `\curlyvee` ``\curlyvee`` `\pm` ``\pm`` `\Cup` ``\Cup`` `\curlywedge` ``\curlywedge`` `\rightthreetimes` ``\rightthreetimes`` `\amalg` ``\amalg`` `\dagger` ``\dagger`` `\rtimes` ``\rtimes`` `\ast` ``\ast`` `\ddagger` ``\ddagger`` `\setminus` ``\setminus`` `\bigcirc` ``\bigcirc`` `\diamond` ``\diamond`` `\smallsetminus` ``\smallsetminus`` `\bigtriangledown` ``\bigtriangledown`` `\div` ``\div`` `\sqcap` ``\sqcap`` `\bigtriangleup` ``\bigtriangleup`` `\divideontimes` ``\divideontimes`` `\sqcup` ``\sqcup`` `\boxdot` ``\boxdot`` `\dotplus` ``\dotplus`` `\star` ``\star`` `\boxminus` ``\boxminus`` `\doublebarwedge` ``\doublebarwedge`` `\times` ``\times`` `\boxplus` ``\boxplus`` `\gtrdot` ``\gtrdot`` `\triangleleft` ``\triangleleft`` `\boxtimes` ``\boxtimes`` `\intercal` ``\intercal`` `\triangleright` ``\triangleright`` `\bullet` ``\bullet`` `\leftthreetimes` ``\leftthreetimes`` `\uplus` ``\uplus`` `\cap` ``\cap`` `\lessdot` ``\lessdot`` `\vee` ``\vee`` `\cdot` ``\cdot`` `\ltimes` ``\ltimes`` `\veebar` ``\veebar`` `\centerdot` ``\centerdot`` `\mp` ``\mp`` `\wedge` ``\wedge`` `\circ` ``\circ`` `\odot` ``\odot`` `\wr` ``\wr`` ================== ==================== ================= =================== ================== ==================== Extensible delimiters --------------------- Unless you indicate otherwise, delimiters in math formulas remain at the standard size regardless of the height of the enclosed material. To get adaptable sizes, use ``\left`` and ``\right`` prefixes, for example `g(A,B,Y) = f \left(A,B,X=h^{[X]}(Y)\right)` or .. math:: a_n = \left(\frac{1}{2}\right)^n Use ``.`` for "empty" delimiters: .. math:: A = \left . \frac{1}{1-n}\, \right |_{n=0}^\infty See also the commands for fixed `delimiter sizes`_ below. The following symbols extend when used with ``\left`` and ``\right``: Pairing delimiters ~~~~~~~~~~~~~~~~~~ .. class:: colwidths-auto =============== ================= ========================= =========================== `( )` ``( )`` `\langle \rangle` ``\langle \rangle`` `[ ]` ``[ ]`` `\lceil \rceil` ``\lceil \rceil`` `\{ \}` ``\{ \}`` `\lfloor \rfloor` ``\lfloor \rfloor`` `\lvert \rvert` ``\lvert \rvert`` `\lgroup \rgroup` ``\lgroup \rgroup`` `\lVert \rVert` ``\lVert \rVert`` `\lmoustache \rmoustache` ``\lmoustache \rmoustache`` =============== ================= ========================= =========================== Nonpairing delimiters ~~~~~~~~~~~~~~~~~~~~~ .. class:: colwidths-auto ==== ====== ============ ============== ============ ============== `|` ``|`` `\vert` ``\vert`` `\arrowvert` ``\arrowvert`` `\|` ``\|`` `\Vert` ``\Vert`` `\Arrowvert` ``\Arrowvert`` `/` ``/`` `\backslash` ``\backslash`` `\bracevert` ``\bracevert`` ==== ====== ============ ============== ============ ============== The use of ``|`` and ``\|`` for pairs of vertical bars may produce incorrect spacing, e.g., ``|k|=|-k|`` produces `|k| = |−k|` and ``|\sin(x)|`` produces `|\sin(x)|`. The pairing delimiters, e.g. `\lvert -k\rvert` and `\lvert\sin(x)\rvert`, prevent this problem. Extensible vertical arrows -------------------------- .. class:: colwidths-auto =============================== ====================================== `\uparrow` ``\uparrow`` `\Uparrow` ``\Uparrow`` `\downarrow` ``\downarrow`` `\Downarrow` ``\Downarrow`` `\updownarrow` ``\updownarrow`` `\Updownarrow` ``\Updownarrow`` =============================== ====================================== Functions (named operators) --------------------------- .. class:: colwidths-auto ========= =========== ========= =========== ============= ================ `\arccos` ``\arccos`` `\gcd` ``\gcd`` `\Pr` ``\Pr`` `\arcsin` ``\arcsin`` `\hom` ``\hom`` `\projlim` ``\projlim`` `\arctan` ``\arctan`` `\inf` ``\inf`` `\sec` ``\sec`` `\arg` ``\arg`` `\injlim` ``\injlim`` `\sin` ``\sin`` `\cos` ``\cos`` `\ker` ``\ker`` `\sinh` ``\sinh`` `\cosh` ``\cosh`` `\lg` ``\lg`` `\sup` ``\sup`` `\cot` ``\cot`` `\lim` ``\lim`` `\tan` ``\tan`` `\coth` ``\coth`` `\liminf` ``\liminf`` `\tanh` ``\tanh`` `\csc` ``\csc`` `\limsup` ``\limsup`` `\varlimsup` ``\varlimsup`` `\deg` ``\deg`` `\ln` ``\ln`` `\varliminf` ``\varliminf`` `\det` ``\det`` `\log` ``\log`` `\varprojlim` ``\varprojlim`` `\dim` ``\dim`` `\max` ``\max`` `\varinjlim` ``\varinjlim`` `\exp` ``\exp`` `\min` ``\min`` ========= =========== ========= =========== ============= ================ Named operators outside the above list can be typeset with ``\operatorname{name}``, e.g. .. math:: \operatorname{sgn}(-3) = -1. .. TODO: \operatorname* for function name with limits. The ``\DeclareMathOperator`` command can only be used in the `LaTeX preamble`_. .. _LaTeX preamble: latex.html#latex-preamble Greek letters ------------- Greek letters that have Latin look-alikes are rarely used in math formulas and not supported by LaTeX. .. class:: colwidths-auto ========== ============ ========== ============ ========== ============ ============== =============== `\Gamma` ``\Gamma`` `\alpha` ``\alpha`` `\mu` ``\mu`` `\omega` ``\omega`` `\Delta` ``\Delta`` `\beta` ``\beta`` `\nu` ``\nu`` `\digamma` ``\digamma`` `\Lambda` ``\Lambda`` `\gamma` ``\gamma`` `\xi` ``\xi`` `\varepsilon` ``\varepsilon`` `\Phi` ``\Phi`` `\delta` ``\delta`` `\pi` ``\pi`` `\varkappa` ``\varkappa`` `\Pi` ``\Pi`` `\epsilon` ``\epsilon`` `\rho` ``\rho`` `\varphi` ``\varphi`` `\Psi` ``\Psi`` `\zeta` ``\zeta`` `\sigma` ``\sigma`` `\varpi` ``\varpi`` `\Sigma` ``\Sigma`` `\eta` ``\eta`` `\tau` ``\tau`` `\varrho` ``\varrho`` `\Theta` ``\Theta`` `\theta` ``\theta`` `\upsilon` ``\upsilon`` `\varsigma` ``\varsigma`` `\Upsilon` ``\Upsilon`` `\iota` ``\iota`` `\phi` ``\phi`` `\vartheta` ``\vartheta`` `\Xi` ``\Xi`` `\kappa` ``\kappa`` `\chi` ``\chi`` `\Omega` ``\Omega`` `\lambda` ``\lambda`` `\psi` ``\psi`` ========== ============ ========== ============ ========== ============ ============== =============== In LaTeX, the default font for capital Greek letters is upright/roman. *Italic* capital Greek letters can be obtained by loading a `package providing the "ISO" math style`__. They are used by default in MathML. Individual Greek italic capitals can also be achieved preceding the letter name with ``var`` like ``\varPhi``: `\varGamma\ \varDelta\ \varLambda\ \varPhi\ \varPi\ \varPsi\ \varSigma\ \varTheta\ \varUpsilon\ \varXi\ \varOmega` __ https://mirrors.ctan.org/macros/latex/contrib/isomath/isomath.html#table-2 Letterlike symbols ------------------ .. class:: colwidths-auto ============= =============== ========== ============ ========== ============ =========== ============= `\forall` ``\forall`` `\aleph` ``\aleph`` `\hbar` ``\hbar`` `\ell` ``\ell`` `\complement` ``\complement`` `\beth` ``\beth`` `\hslash` ``\hslash`` `\wp` ``\wp`` `\exists` ``\exists`` `\gimel` ``\gimel`` `\Im` ``\Im`` `\Re` ``\Re`` `\Finv` ``\Finv`` `\daleth` ``\daleth`` `\imath` ``\imath`` `\circledR` ``\circledR`` `\Game` ``\Game`` `\partial` ``\partial`` `\jmath` ``\jmath`` `\circledS` ``\circledS`` `\mho` ``\mho`` `\eth` ``\eth`` `\Bbbk` ``\Bbbk`` ============= =============== ========== ============ ========== ============ =========== ============= .. _math alphabet: Math alphabets -------------- The TeX *math alphabet* macros are intended for mathematical variables where style variations are important semantically. They style letters and numbers with a combination of font attributes (shape, weight, family) --- non-alphanumerical symbols, function names, and mathematical text are left unchanged. MathML uses the *mathvariant* `style attribute`_ or pre-styled characters from the `Mathematical Alphanumeric Symbols`_ Unicode block. .. class:: colwidths-auto =============== ============================ ========================== command example result =============== ============================ ========================== ``\mathrm`` ``s_\mathrm{out}`` `s_\mathrm{out}` ``\mathbf`` ``\mathbf{r}^2=x^2+y^2`` `\mathbf{r}^2=x^2+y^2` ``\mathit`` ``\mathit{\sin\Gamma}`` `\mathit{\sin\Gamma}` ``\mathcal`` ``\mathcal{F}f(x)`` `\mathcal{F}f(x)` ``\mathbb`` ``\mathbb{R \subset C}`` `\mathbb{R \subset C}` ``\mathfrak`` ``\mathfrak{a+b}`` `\mathfrak{a+b}` ``\mathsf`` ``\mathsf x`` `\mathsf x` ``\mathtt`` ``\mathtt{0.12}`` `\mathtt{0.12}` =============== ============================ ========================== The set of characters in a given "math alphabet" varies. LaTeX may produce garbage for unsupported characters. Additional math alphabets are defined in LaTeX packages, e.g., * ``\mathbfit`` from isomath_ allows vector symbols in line with the International Standard [ISO-80000-2]_. E.g., ``\mathbfit{r}^2=x^2+y^2`` becomes `\mathbfit{r}^2=x^2+y^2`. * Several packages, e.g. mathrsfs_, define ``\mathscr`` that selects a differently shaped "script" alphabet. The listing below shows the characters supported by Unicode and Docutils with math_output_ MathML. [#italic-digits]_ default: `{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz\ ı\jmath}` `{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖϜϝ\ \partial∇}` `{0123456789}` mathrm: `\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz\ ı\jmath}` `\mathrm{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖϜϝ\ \partial∇}` `\mathrm{0123456789}` mathbf: `\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathbf{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖϜϝ\ \partial∇}` `\mathbf{0123456789}` mathit: `\mathit{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz\ ı\jmath}` `\mathit{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖ\ \partial∇}` `\mathit{0123456789}` [#italic-digits]_ mathbfit: `\mathbfit{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathbfit{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖ\ \partial∇}` mathcal: `\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` mathscr: `\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` mathbb: `\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathbb{ΓΠΣ\ γπ}` `\mathbb{0123456789}` mathfrak: `\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` mathsf: `\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathsf{0123456789}` mathsfit: `\mathsfit{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` mathsfbfit: `\mathsfbfit{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathsfbfit{ΓΔΘΛΞΠΣΥΦΨΩ\ αβγδεζηθικλμνξπρςστυφχψω\ ϵϑϕϰϱϖ\ \partial∇}` mathtt: `\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ\ abcdefghijklmnopqrstuvwxyz}` `\mathtt{0123456789}` .. [#italic-digits] Italic digits are not defined in Unicode but work in LaTeX. In contrast to the *math alphabet* selectors, ``\boldsymbol`` only changes the *font weight*. It can be used to get a bold version of any mathematical symbol: .. math:: V_i x \pm \cos(\alpha) \approx 3\Gamma \quad \forall x\in\mathbb{R} \boldsymbol{V_i x \pm \cos(\alpha) \approx 3\Gamma \quad \forall x\in\mathbb{R}} It is usually ill-advised to apply ``\boldsymbol`` to more than one symbol at a time. .. _style attribute: https://www.w3.org/TR/mathml4/#presm_commatt .. _Mathematical Alphanumeric Symbols: https://en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols .. _isomath: https://www.ctan.org/pkg/isomath .. _mathrsfs: https://www.ctan.org/pkg/mathrsfs .. _unicode-math: https://www.ctan.org/pkg/unicode-math Miscellaneous symbols --------------------- .. class:: colwidths-auto ==================== ====================== ================ ================== ================= =================== `\#` ``\#`` `\clubsuit` ``\clubsuit`` `\neg` ``\neg`` `\&` ``\&`` `\diamondsuit` ``\diamondsuit`` `\nexists` ``\nexists`` `\angle` ``\angle`` `\emptyset` ``\emptyset`` `\prime` ``\prime`` `\backprime` ``\backprime`` `\exists` ``\exists`` `\sharp` ``\sharp`` `\bigstar` ``\bigstar`` `\flat` ``\flat`` `\spadesuit` ``\spadesuit`` `\blacklozenge` ``\blacklozenge`` `\forall` ``\forall`` `\sphericalangle` ``\sphericalangle`` `\blacksquare` ``\blacksquare`` `\heartsuit` ``\heartsuit`` `\square` ``\square`` `\blacktriangle` ``\blacktriangle`` `\infty` ``\infty`` `\surd` ``\surd`` `\blacktriangledown` ``\blacktriangledown`` `\lozenge` ``\lozenge`` `\top` ``\top`` `\bot` ``\bot`` `\measuredangle` ``\measuredangle`` `\triangle` ``\triangle`` `\diagdown` ``\diagdown`` `\nabla` ``\nabla`` `\triangledown` ``\triangledown`` `\diagup` ``\diagup`` `\natural` ``\natural`` `\varnothing` ``\varnothing`` ==================== ====================== ================ ================== ================= =================== Punctuation ----------- .. class:: colwidths-auto === ===== ======== =============== ======== ========== `.` ``.`` `!` ``!`` `\vdots` ``\vdots`` `/` ``/`` `?` ``?`` `\dotsb` ``\dotsb`` `|` ``|`` `\colon` ``\colon`` [#]_ `\dotsc` ``\dotsc`` `'` ``'`` `\cdots` ``\cdots`` `\dotsi` ``\dotsi`` `;` ``;`` `\ddots` ``\ddots`` `\dotsm` ``\dotsm`` `:` ``:`` `\ldots` ``\ldots`` `\dotso` ``\dotso`` === ===== ======== =============== ======== ========== .. [#] Punctuation (not ratio): Compare spacing in `a\colon b\to c` to `a:b = c`. Relation symbols ---------------- Arrows ~~~~~~ .. class:: colwidths-auto ====================== ======================== ===================== ======================= `\circlearrowleft` ``\circlearrowleft`` `\circlearrowright` ``\circlearrowright`` `\curvearrowleft` ``\curvearrowleft`` `\curvearrowright` ``\curvearrowright`` `\hookleftarrow` ``\hookleftarrow`` `\hookrightarrow` ``\hookrightarrow`` `\leftarrow` ``\leftarrow`` `\rightarrow` ``\rightarrow`` `\Leftarrow` ``\Leftarrow`` `\Rightarrow` ``\Rightarrow`` `\leftarrowtail` ``\leftarrowtail`` `\rightarrowtail` ``\rightarrowtail`` `\leftharpoondown` ``\leftharpoondown`` `\rightharpoondown` ``\rightharpoondown`` `\leftharpoonup` ``\leftharpoonup`` `\rightharpoonup` ``\rightharpoonup`` `\leftleftarrows` ``\leftleftarrows`` `\rightrightarrows` ``\rightrightarrows`` `\leftrightarrow` ``\leftrightarrow`` `\Leftrightarrow` ``\Leftrightarrow`` `\leftrightarrows` ``\leftrightarrows`` `\rightleftarrows` ``\rightleftarrows`` `\leftrightharpoons` ``\leftrightharpoons`` `\rightleftharpoons` ``\rightleftharpoons`` `\leftrightsquigarrow` ``\leftrightsquigarrow`` `\rightsquigarrow` ``\rightsquigarrow`` `\Lleftarrow` ``\Lleftarrow`` `\Rrightarrow` ``\Rrightarrow`` `\longleftarrow` ``\longleftarrow`` `\longrightarrow` ``\longrightarrow`` `\Longleftarrow` ``\Longleftarrow`` `\Longrightarrow` ``\Longrightarrow`` `\longleftrightarrow` ``\longleftrightarrow`` `\Longleftrightarrow` ``\Longleftrightarrow`` `\looparrowleft` ``\looparrowleft`` `\looparrowright` ``\looparrowright`` `\Lsh` ``\Lsh`` `\Rsh` ``\Rsh`` `\mapsto` ``\mapsto`` `\longmapsto` ``\longmapsto`` `\multimap` ``\multimap`` `\nleftarrow` ``\nleftarrow`` `\nrightarrow` ``\nrightarrow`` `\nLeftarrow` ``\nLeftarrow`` `\nRightarrow` ``\nRightarrow`` `\nleftrightarrow` ``\nleftrightarrow`` `\nLeftrightarrow` ``\nLeftrightarrow`` `\nwarrow` ``\nwarrow`` `\nearrow` ``\nearrow`` `\swarrow` ``\swarrow`` `\searrow` ``\searrow`` `\twoheadleftarrow` ``\twoheadleftarrow`` `\twoheadrightarrow` ``\twoheadrightarrow`` `\upharpoonleft` ``\upharpoonleft`` `\upharpoonright` ``\upharpoonright`` `\downharpoonleft` ``\downharpoonleft`` `\downharpoonright` ``\downharpoonright`` `\upuparrows` ``\upuparrows`` `\downdownarrows` ``\downdownarrows`` ====================== ======================== ===================== ======================= Synonyms: `\gets` ``\gets``, `\to` ``\to``, `\restriction` ``\restriction``. Comparison ~~~~~~~~~~ .. class:: colwidths-auto ================ ================== ============= =============== ============= =============== =============== ================= `<` ``<`` `\geq` ``\geq`` `\ll` ``\ll`` `\prec` ``\prec`` `=` ``=`` `\geqq` ``\geqq`` `\lll` ``\lll`` `\precapprox` ``\precapprox`` `>` ``>`` `\geqslant` ``\geqslant`` `\lnapprox` ``\lnapprox`` `\preccurlyeq` ``\preccurlyeq`` `\approx` ``\approx`` `\gg` ``\gg`` `\lneq` ``\lneq`` `\preceq` ``\preceq`` `\approxeq` ``\approxeq`` `\ggg` ``\ggg`` `\lneqq` ``\lneqq`` `\precnapprox` ``\precnapprox`` `\asymp` ``\asymp`` `\gnapprox` ``\gnapprox`` `\lnsim` ``\lnsim`` `\precneqq` ``\precneqq`` `\backsim` ``\backsim`` `\gneq` ``\gneq`` `\ncong` ``\ncong`` `\precnsim` ``\precnsim`` `\backsimeq` ``\backsimeq`` `\gneqq` ``\gneqq`` `\neq` ``\neq`` `\precsim` ``\precsim`` `\bumpeq` ``\bumpeq`` `\gnsim` ``\gnsim`` `\ngeq` ``\ngeq`` `\risingdotseq` ``\risingdotseq`` `\Bumpeq` ``\Bumpeq`` `\gtrapprox` ``\gtrapprox`` `\ngeqq` ``\ngeqq`` `\sim` ``\sim`` `\circeq` ``\circeq`` `\gtreqless` ``\gtreqless`` `\ngeqslant` ``\ngeqslant`` `\simeq` ``\simeq`` `\cong` ``\cong`` `\gtreqqless` ``\gtreqqless`` `\ngtr` ``\ngtr`` `\succ` ``\succ`` `\curlyeqprec` ``\curlyeqprec`` `\gtrless` ``\gtrless`` `\nleq` ``\nleq`` `\succapprox` ``\succapprox`` `\curlyeqsucc` ``\curlyeqsucc`` `\gtrsim` ``\gtrsim`` `\nleqq` ``\nleqq`` `\succcurlyeq` ``\succcurlyeq`` `\doteq` ``\doteq`` `\leq` ``\leq`` `\nleqslant` ``\nleqslant`` `\succeq` ``\succeq`` `\doteqdot` ``\doteqdot`` `\leqq` ``\leqq`` `\nless` ``\nless`` `\succnapprox` ``\succnapprox`` `\eqcirc` ``\eqcirc`` `\leqslant` ``\leqslant`` `\nprec` ``\nprec`` `\succneqq` ``\succneqq`` `\eqsim` ``\eqsim`` `\lessapprox` ``\lessapprox`` `\npreceq` ``\npreceq`` `\succnsim` ``\succnsim`` `\eqslantgtr` ``\eqslantgtr`` `\lesseqgtr` ``\lesseqgtr`` `\nsim` ``\nsim`` `\succsim` ``\succsim`` `\eqslantless` ``\eqslantless`` `\lesseqqgtr` ``\lesseqqgtr`` `\nsucc` ``\nsucc`` `\thickapprox` ``\thickapprox`` `\equiv` ``\equiv`` `\lessgtr` ``\lessgtr`` `\nsucceq` ``\nsucceq`` `\thicksim` ``\thicksim`` `\fallingdotseq` ``\fallingdotseq`` `\lesssim` ``\lesssim`` `\triangleq` ``\triangleq`` ================ ================== ============= =============== ============= =============== =============== ================= The commands ``\lvertneqq`` and ``\gvertneqq`` are not supported with MathML output, as there is no corresponding Unicode character. Synonyms: `\ne` ``\ne``, `\le` ``\le``, `\ge` ``\ge``, `\Doteq` ``\Doteq``, `\llless` ``\llless``, `\gggtr` ``\gggtr``. Symbols can be negated prepending ``\not``, e.g. `\not=` ``\not=``, `\not\equiv` ``\not\equiv``, `\not\gtrless` ``\not\gtrless``, `\not\lessgtr` ``\not\lessgtr``. Miscellaneous relations ~~~~~~~~~~~~~~~~~~~~~~~ .. class:: colwidths-auto ===================== ======================= =================== ===================== =================== ===================== `\backepsilon` ``\backepsilon`` `\ntrianglelefteq` ``\ntrianglelefteq`` `\subseteq` ``\subseteq`` `\because` ``\because`` `\ntriangleright` ``\ntriangleright`` `\subseteqq` ``\subseteqq`` `\between` ``\between`` `\ntrianglerighteq` ``\ntrianglerighteq`` `\subsetneq` ``\subsetneq`` `\blacktriangleleft` ``\blacktriangleleft`` `\nvdash` ``\nvdash`` `\subsetneqq` ``\subsetneqq`` `\blacktriangleright` ``\blacktriangleright`` `\nVdash` ``\nVdash`` `\supset` ``\supset`` `\bowtie` ``\bowtie`` `\nvDash` ``\nvDash`` `\Supset` ``\Supset`` `\dashv` ``\dashv`` `\nVDash` ``\nVDash`` `\supseteq` ``\supseteq`` `\frown` ``\frown`` `\parallel` ``\parallel`` `\supseteqq` ``\supseteqq`` `\in` ``\in`` `\perp` ``\perp`` `\supsetneq` ``\supsetneq`` `\mid` ``\mid`` `\pitchfork` ``\pitchfork`` `\supsetneqq` ``\supsetneqq`` `\models` ``\models`` `\propto` ``\propto`` `\therefore` ``\therefore`` `\ni` ``\ni`` `\shortmid` ``\shortmid`` `\trianglelefteq` ``\trianglelefteq`` `\nmid` ``\nmid`` `\shortparallel` ``\shortparallel`` `\trianglerighteq` ``\trianglerighteq`` `\notin` ``\notin`` `\smallfrown` ``\smallfrown`` `\varpropto` ``\varpropto`` `\nparallel` ``\nparallel`` `\smallsmile` ``\smallsmile`` `\vartriangle` ``\vartriangle`` `\nshortmid` ``\nshortmid`` `\smile` ``\smile`` `\vartriangleleft` ``\vartriangleleft`` `\nshortparallel` ``\nshortparallel`` `\sqsubset` ``\sqsubset`` `\vartriangleright` ``\vartriangleright`` `\nsubseteq` ``\nsubseteq`` `\sqsubseteq` ``\sqsubseteq`` `\vdash` ``\vdash`` `\nsubseteqq` ``\nsubseteqq`` `\sqsupset` ``\sqsupset`` `\Vdash` ``\Vdash`` `\nsupseteq` ``\nsupseteq`` `\sqsupseteq` ``\sqsupseteq`` `\vDash` ``\vDash`` `\nsupseteqq` ``\nsupseteqq`` `\subset` ``\subset`` `\Vvdash` ``\Vvdash`` `\ntriangleleft` ``\ntriangleleft`` `\Subset` ``\Subset`` ===================== ======================= =================== ===================== =================== ===================== Synonyms: `\owns` ``\owns``. Symbols can be negated prepending ``\not``, e.g. `\not\in` ``\not\in``, `\not\ni` ``\not\ni``. The commands ``\varsubsetneq``, ``\varsubsetneqq``, ``\varsupsetneq``, and ``\varsupsetneqq`` are not supported with MathML output as there is no corresponding Unicode character. Variable-sized operators ------------------------ .. class:: colwidths-auto ========================= ========================= ========================= =========================== `\sum` ``\sum`` `\prod` ``\prod`` `\bigcap` ``\bigcap`` `\bigodot` ``\bigodot`` `\int` ``\int`` `\coprod` ``\coprod`` `\bigcup` ``\bigcup`` `\bigoplus` ``\bigoplus`` `\oint` ``\oint`` `\bigwedge` ``\bigwedge`` `\biguplus` ``\biguplus`` `\bigotimes` ``\bigotimes`` `\smallint` ``\smallint`` `\bigvee` ``\bigvee`` `\bigsqcup` ``\bigsqcup`` ========================= ========================= ========================= =========================== Larger symbols are used in displayed formulas, sum-like symbols have indices above/below the symbol: .. math:: \sum_{n=1}^N a_n \qquad \int_0^1f(x)\,dx \qquad \prod_{i=1}^{10} b_i \ldots Notations ========= Top and bottom embellishments ----------------------------- See `Accents and embellishments`_. Extensible arrows ----------------- \xleftarrow and \xrightarrow produce arrows that extend automatically to accommodate unusually wide subscripts or superscripts. These commands take one optional argument (the subscript) and one mandatory argument (the superscript, possibly empty):: A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C results in .. math:: A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C Affixing symbols to other symbols --------------------------------- In addition to the standard `accents and embellishments`_, other symbols can be placed above or below a base symbol with the ``\overset`` and ``\underset`` commands. The symbol is set in "scriptstyle" (smaller font size). For example, writing ``\overset{*}{X}`` becomes `\overset{*}{X}` and ``\underset{+}{M}`` becomes `\underset{+}{M}`. Matrices -------- The ``matrix`` and ``cases`` environments can also contain ``\\`` and ``&``:: .. math:: \left ( \begin{matrix} a & b \\ c & d \end{matrix}\right) Result: .. math:: \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right) The environments ``pmatrix``, ``bmatrix``, ``Bmatrix``, ``vmatrix``, and ``Vmatrix`` have (respectively) ( ), [ ], { }, \| \|, and `\Vert\ \Vert` delimiters built in, e.g. .. math:: \begin{pmatrix} a & b \\ c & d \end{pmatrix} \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \qquad \begin{Vmatrix} a & b \\ c & d \end{Vmatrix} To produce a small matrix suitable for use in text, there is a ``smallmatrix`` environment `\bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr)` that comes closer to fitting within a single text line than a normal matrix. For piecewise function definitions there is a ``cases`` environment: .. math:: \mathrm{sgn}(x) = \begin{cases} -1 & x<0\\ \phantom{-}1 & x>0 \end{cases} Spacing commands ---------------- Horizontal spacing of elements can be controlled with the following commands: .. class:: colwidths-auto ====================== ======== ===================== =================== :m:`3\qquad 4` ``3\qquad 4`` = 2em :m:`3\quad 4` ``3\quad 4`` = 1em :m:`3~4` ``3~4`` ``3\nobreakspace 4`` :m:`3\ 4` ``3\ 4`` escaped space :m:`3\;4` ``3\;4`` ``3\thickspace 4`` :m:`3\:4` ``3\:4`` ``3\medspace 4`` :m:`3\,4` ``3\,4`` ``3\thinspace 4`` :m:`3 4` ``3 4`` regular space [#]_ :m:`3\!4` ``3\!4`` ``3\negthinspace 4`` negative space [#]_ :m:`3\negmedspace 4` ``3\negmedspace 4`` :m:`3\negthickspace 4` ``3\negthickspace 4`` `3\hspace{1ex}4` ``3\hspace{1ex}4`` custom length `3\mspace{20mu}4` ``3\mspace{20mu}4`` custom length [#]_ ====================== ======== ===================== =================== .. [#] Whitespace characters are ignored in LaTeX math mode. .. [#] Negative spacing does not work with MathML (last tested in Firefox 115). .. [#] In LaTeX, the unit must be 'mu' (1 mu = 1/18em). There are also three commands that leave a space equal to the height and width of its argument. For example ``\phantom{XXX}`` results in space as wide and high as three X’s: .. math:: \frac{\phantom{XXX}+1}{XXX-1} The commands ``\hphantom`` and ``\vphantom`` insert space with the width or height of the argument. They are not supported with `math_output`_ MathML. Modular arithmetic and modulo operation --------------------------------------- The commands ``\bmod``, ``\pmod``, ``\mod``, and ``\pod`` deal with the special spacing conventions of the “mod” notation. [#]_ .. class:: colwidths-auto ========= =========================== ========================= command example result ========= =========================== ========================= ``\bmod`` ``\gcd(n,m \bmod n)`` `\gcd(n,m \bmod n)` ``\pmod`` ``x\equiv y \pmod b`` `x\equiv y \pmod b` ``\mod`` ``x\equiv y \mod c`` `x\equiv y \mod c` ``\pod`` ``x\equiv y \pod d`` `x\equiv y \pod d` .. ``\operatorname{mod}(m,n)`` `\operatorname{mod}(m,n)` ========= =========================== ========================= .. [#] Currently `not supported`__ by the "HTML" math_output_ option of the HTML writer. __ https://sourceforge.net/p/docutils/feature-requests/93/ Roots ----- .. class:: colwidths-auto ========= ==================== ================== command example result ========= ==================== ================== ``\sqrt`` ``\sqrt{x^2-1}`` `\sqrt{x^2-1}` .. ``\sqrt[3n]{x^2-1}`` `\sqrt[3n]{x^2-1}` .. ``\sqrt\frac{1}{2}`` `\sqrt\frac{1}{2}` ========= ==================== ================== Boxed formulas -------------- The command ``\boxed`` puts a box around its argument: .. math:: \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))} Fractions and related constructions =================================== The ``\frac`` command takes two ar guments, numerator and denominator, and typesets them in normal fraction form. For example, ``U = \frac{R}{I}`` produces `U = \frac{R}{I}`. Use ``\dfrac`` or ``\tfrac`` to force text style and display style respectively. .. math:: \frac{x+1}{x-1} \quad \dfrac{x+1}{x-1} \quad \tfrac{x+1}{x-1} and in text: `\frac{x+1}{x-1}`, `\dfrac{x+1}{x-1}`, `\tfrac{x+1}{x-1}`. For binomial expressions such as `\binom{n}{k}`, there are ``\binom``, ``\dbinom`` and ``\tbinom`` commands:: 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} prints .. math:: 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2} The ``\cfrac`` command for continued fractions uses displaystyle and padding for sub-fractions: .. math:: \frac{\pi}{4} = 1 + \cfrac{1^2}{ 2 + \cfrac{3^2}{ 2 + \cfrac{5^2}{ 2 + \cfrac{7^2}{2 + \cdots} }}} \qquad \text{vs.}\qquad \frac{\pi}{4} = 1 + \frac{1^2}{ 2 + \frac{3^2}{ 2 + \frac{5^2}{ 2 + \frac{7^2}{2 + \cdots} }}} The optional argument ``[l]`` or ``[r]`` for left or right placement of the numerator is `not supported by MathML Core`__: .. math:: \cfrac[l]{x}{x-1} \quad \cfrac{x}{x-1} \quad \cfrac[r]{x}{x-1} __ https://github.com/w3c/mathml/issues/30 Delimiter sizes =============== Besides the automatic scaling of `extensible delimiters`_ with ``\left`` and ``\right``, there are four commands to manually select delimiters of fixed size: .. class:: colwidths-auto ========= ============== ============== ============== ============== =============== =============== Sizing no ``\left`` ``\bigl`` ``\Bigl`` ``\biggl`` ``\Biggl`` command ``\right`` ``\bigr`` ``\Bigr`` ``\biggr`` ``\Biggr`` --------- -------------- -------------- -------------- -------------- --------------- --------------- Result `\displaystyle `\displaystyle `\displaystyle `\displaystyle `\displaystyle `\displaystyle (b) \left(b\right) \bigl(b\bigr) \Bigl(b\Bigr) \biggl(b\biggr) \Biggl(b\Biggr) (\frac{c}{d})` \left(\frac{c} \bigl(\frac{c} \Bigl(\frac{c} \biggl(\frac{c} \Biggl(\frac{c} {d}\right)` {d}\bigr)` {d}\Bigr)` {d}\biggr)` {d}\Biggr)` ========= ============== ============== ============== ============== =============== =============== There are two or three situations where the delimiter size is commonly adjusted using these commands: The first kind of adjustment is done for cumulative operators with limits, such as summation signs. With ``\left`` and ``\right`` the delimiters usually turn out larger than necessary, and using the ``Big`` or ``bigg`` sizes instead gives better results: .. math:: \left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p} \text{ versus } \biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p} The second kind of situation is clustered pairs of delimiters, where \left and \right make them all the same size (because that is adequate to cover the encompassed material), but what you really want is to make some of the delimiters slightly larger to make the nesting easier to see. .. math:: \left((a_1 b_1) - (a_2 b_2)\right) \left((a_2 b_1) + (a_1 b_2)\right) \quad\text{versus}\quad \bigl((a_1 b_1) - (a_2 b_2)\bigr) \bigl((a_2 b_1) + (a_1 b_2)\bigr) The third kind of situation is a slightly oversize object in running text, such as `\left|\frac{b'}{d'}\right|` where the delimiters produced by ``\left`` and ``\right`` cause too much line spreading. [#]_ In that case ``\bigl`` and ``\bigr`` can be used to produce delimiters that are larger than the base size but still able to fit within the normal line spacing: `\bigl|\frac{b'}{d'}\bigr|`. .. [#] With MathML, an example would be parentheses around a ``smallmatrix`` environment `\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)` vs. `\Bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\Bigr)`. Text ==== The main use of the command ``\text`` is for words or phrases in a display. It is similar to ``\mbox`` in its effects but, unlike ``\mbox``, automatically produces subscript-size text if used in a subscript, ``k_{\text{B}}T`` becomes `k_{\text{B}}T`. Whitespace is kept inside the argument: .. Math:: f_{[x_{i-1},x_i]} \text{ is monotonic for } i = 1,\,…,\,c+1 The text may contain math commands wrapped in ``$`` signs, e.g. .. math:: (-1)^{n_i} = \begin{cases} -1 \quad \text{if $n_i$ is odd,} \\ +1 \quad \text{if $n_i$ is even.} \end{cases} .. TODO ignore {}, handle text-mode commands Integrals and sums ================== The limits on integrals, sums, and similar symbols are placed either to the side of or above and below the base symbol, depending on convention and context. In inline formulas and fractions, the limits on sums, and similar symbols like .. math:: \lim_{n\to\infty} \sum_1^n \frac{1}{n} move to index positions: `\lim_{n\to\infty} \sum_1^n \frac{1}{n}`. Altering the placement of limits -------------------------------- The commands ``\intop`` and ``\ointop`` produce integral signs with limits as in sums and similar: `\intop_0^1`, `\ointop_c` and .. math:: \intop_0^1 \quad \ointop_c \quad \text{vs.} \quad \int^1_0 \quad \oint_c The commands ``\limits`` and ``\nolimits`` override the default placement of the limits for any operator; ``\displaylimits`` forces standard positioning as for the \sum command. They should follow immediately after the operator to which they apply. Compare the same term with default positions, ``\limits``, and ``\nolimits`` in inline and display mode: `\lim_{x\to0}f(x)`, `\lim\limits_{x\to0}f(x)`, `\lim\nolimits_{x\to0}f(x)`, vs. .. math:: \lim_{x\to0}f(x), \quad \lim\limits_{x\to0}f(x) \quad \lim\nolimits_{x\to0}f(x). .. TODO: \substack .. TODO: \sideset Changing the size of elements in a formula ========================================== The declarations [#]_ ``\displaystyle``, ``\textstyle``, ``\scriptstyle``, and ``\scriptscriptstyle``, select a symbol size and spacing that would be applied in display math, inline math, first-order subscript, or second-order subscript, respectively even when the current context would normally yield some other size. For example ``:math:`\displaystyle \sum_{n=0}^\infty \frac{1}{n}``` is printed as `\displaystyle \sum_{n=0}^\infty \frac{1}{n}` rather than `\sum_{n=0}^\infty \frac{1}{n}` and :: \frac{\scriptstyle\sum_{n > 0} z^n} {\displaystyle\prod_{1\leq k\leq n} (1-q^k)} yields .. math:: \frac{\scriptstyle\sum_{n > 0} z^n} {\displaystyle\prod_{1\leq k\leq n} (1-q^k)} \text{ instead of the default } \frac{\sum_{n > 0} z^n} {\prod_{1\leq k\leq n} (1-q^k)}. .. [#] "Declarations" are commands that affect processing of the current "group". In particular, notice where the braces fall that delimit the effect of the command: Right: ``{\displaystyle ...}`` Wrong: ``\displaystyle{...}``. With math_output_ MathML, the declaration must be the first element after the opening bracket. .. [ISO-80000-2] :title:`Quantities and units – Part 2: Mathematical signs and symbols to be used in the natural sciences and technology`: http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=31887.