Mathematics - A Personal Blog https://farid.partonia.ir/ en Mathematics - A Personal Blog[shortrss] LaTeX mathematic cheat sheet https://farid.partonia.ir/index.php?newsid=17 https://farid.partonia.ir/index.php?newsid=17

A complete set of tables for writing in LaTeX which comprises:

  • Accents/diacritics
  • Standard functions
  • Modular arithmetic
  • Derivatives
  • Sets
  • Operators
  • Logic
  • Root
  • Relations
  • Geometric
  • Arrows
  • Special
  • Subscripts, superscripts, integrals
  • Fractions, matrices, multi lines
  • Parenthesizing big expressions, brackets, bars
  • Alphabets
 [allow-turbo]Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


]]>[/allow-turbo] Programming, Mathematics FariD Sun, 09 Jan 2022 15:55:28 +0330 [/shortrss] [fullrss] LaTeX mathematic cheat sheet https://farid.partonia.ir/index.php?newsid=17 https://farid.partonia.ir/index.php?newsid=17 FariD Sun, 09 Jan 2022 15:55:28 +0330 A complete set of tables for writing in LaTeX which comprises:

  • Accents/diacritics
  • Standard functions
  • Modular arithmetic
  • Derivatives
  • Sets
  • Operators
  • Logic
  • Root
  • Relations
  • Geometric
  • Arrows
  • Special
  • Subscripts, superscripts, integrals
  • Fractions, matrices, multi lines
  • Parenthesizing big expressions, brackets, bars
  • Alphabets
]]> [allow-turbo]Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


]]>[/allow-turbo] [allow-dzen]Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


]]>[/allow-dzen] [/fullrss] [yandexrss] LaTeX mathematic cheat sheet https://farid.partonia.ir/index.php?newsid=17

A complete set of tables for writing in LaTeX which comprises:

  • Accents/diacritics
  • Standard functions
  • Modular arithmetic
  • Derivatives
  • Sets
  • Operators
  • Logic
  • Root
  • Relations
  • Geometric
  • Arrows
  • Special
  • Subscripts, superscripts, integrals
  • Fractions, matrices, multi lines
  • Parenthesizing big expressions, brackets, bars
  • Alphabets
 Programming, Mathematics Sun, 09 Jan 2022 15:55:28 +0330

Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


 [allow-turbo]Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


]]>[/allow-turbo] [allow-dzen]Practically, LaTeX is the standard typesetting system for scientific writing. Most of the well-written equations that appeared in books and around the web are written using LaTeX.

Accents/diacritics

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

 {\acute {a}}{\grave {a}}{\hat {a}}{\tilde {a}}{\breve {a}}\,

\check{a} \bar{a} \ddot{a} \dot{a}

 {\check {a}}{\bar {a}}{\ddot {a}}{\dot {a}}

Standard functions

\sin a \cos b \tan c

 \sin a\cos b\tan c

\sec d \csc e \cot f

 \sec d\csc e\cot f\,

\arcsin h \arccos i \arctan j

 \arcsin h\arccos i\arctan j\,

\sinh k \cosh l \tanh m \coth n

 \sinh k\cosh l\tanh m\coth n

\operatorname{sh}o\, \operatorname{ch}p\, \operatorname{th}q

 \operatorname {sh} o\,\operatorname {ch} p\,\operatorname {th} q

\operatorname{arsinh}r\, \operatorname{arcosh}s\, \operatorname{artanh}t

 \operatorname {arsinh} r\,\operatorname {arcosh} s\,\operatorname {artanh} t

\lim u \limsup v \liminf w \min x \max y

 \lim u\limsup v\liminf w\min x\max y

\inf z \sup a \exp b \ln c \lg d \log e \log_{10} f \ker g

 \inf z\sup a\exp b\ln c\lg d\log e\log _{10}f\ker g

\deg h \gcd i \Pr j \det k \hom l \arg m \dim n

 \deg h\gcd i\Pr j\det k\hom l\arg m\dim n


Modular arithmetic

s_k \equiv 0 \pmod{m}

 s_{k}\equiv 0{\pmod {m}}\,

a\, \bmod\, b

 a\,{\bmod {\,}}b\,

Derivatives

\nabla\, \partial x\, dx\, \dot x\, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}, {\partial x_1\,\partial x_2}

 \nabla \,\partial x\,dx\,{\dot {x}}\,{\ddot {y}}\,dy/dx\,{\frac {dy}{dx}}\,{\frac {\partial ^{2}y}{\partial x_{1}\,\partial x_{2}}}

Sets

\forall \exists \empty \emptyset \varnothing

 \forall \exists \emptyset \emptyset \varnothing \,

\in \ni \not\in \notin \not\ni \subset \subseteq \supset \supseteq

 {\displaystyle \in \ni \not \in \notin \not \ni \subset \subseteq \supset \supseteq \,}

\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus

 \cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus \,

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup

 \sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup \,

Operators

+ \oplus \bigoplus \pm \mp -

 +\oplus \bigoplus \pm \mp -\,

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

 \times \otimes \bigotimes \cdot \circ \bullet \bigodot \,

\star */ \div \frac{1}{2}

 \star */\div {\frac {1}{2}}\,

Logic

\land (or \and) \wedge \bigwedge \bar{q} \to p

 \land \wedge \bigwedge {\bar {q}}\to p\,

\lor \vee \bigvee \lnot \neg q \And

 \lor \vee \bigvee \lnot \neg q\And \,

Root

\sqrt{2} \sqrt[n]{x}

 {\sqrt {2}}{\sqrt[{n}]{x}}\,

Relations

\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}

 \sim \approx \simeq \cong {\dot {=}}{\overset {\underset {\mathrm {def} }{}}{=}}\,

< \le \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto

 <\leq \ll \gg \geq >\equiv \not \equiv \neq {\mbox{or}}\neq \propto \,

\lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

 \lessapprox \lesssim \eqslantless \leqslant \leqq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox

Geometric

\Diamond \Box \triangle \angle \perp \mid \nmid \| 45^\circ

 \Diamond \,\Box \,\triangle \,\angle \perp \,\mid \;\nmid \,\|45^{\circ }\,

Arrows

\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow

 \leftarrow \rightarrow \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow \,

\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow

(or \impliedby) \Longrightarrow (or \implies) \Longleftrightarrow (or \iff)

 \Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

 \uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow

\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons

 \rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons \,

\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow

\rightarrowtail \looparrowright

 \curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright \,

\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow

\leftarrowtail \looparrowleft

 \curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft \,

\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow

 \mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \,

Special

\And \eth \S \P \% \dagger \ddagger \ldots \cdots \colon

 {\displaystyle \And \eth \S \P \%\dagger \ddagger \ldots \cdots \colon \,}

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

 \smile \frown \wr \triangleleft \triangleright \infty \bot \top \,

\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar

 \vdash \vDash \Vdash \models \lVert \rVert \imath \hbar \,

\ell \mho \Finv \Re \Im \wp \complement

 \ell \mho \Finv \Re \Im \wp \complement \,

\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp

 \diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp \,

USubscripts, superscripts, integrals

Feature Syntax How it looks rendered
Superscript

a^2

 a^{2}
Subscript

a_2

 a_{2}
Grouping

a^{2+2}

 a^{2+2}

a_{i,j}

 a_{i,j}
Combining sub & super without and with horizontal separation

x_2^3

 x_{2}^{3}

{x_2}^3

 {x_{2}}^{3}
Super super

10^{10^{8}}

 10^{10^{8}}
Preceding and/or Additional sub & super

_nP_k

 _{n}P_{k}

\sideset{_1^2}{_3^4}\prod_a^b

 \sideset {_{1}^{2}}{_{3}^{4}}\prod _{a}^{b}

{}_1^2\!\Omega_3^4

 {}_{1}^{2}\!\Omega _{3}^{4}
Stacking

\overset{\alpha}{\omega}

 {\overset {\alpha }{\omega }}

\underset{\alpha}{\omega}

 {\underset {\alpha }{\omega }}

\overset{\alpha}{\underset{\gamma}{\omega}}

 {\overset {\alpha }{\underset {\gamma }{\omega }}}

\stackrel{\alpha}{\omega}

 {\stackrel {\alpha }{\omega }}
Derivatives

x', y'', f', f''

 x',y'',f',f''

x^\prime, y^{\prime\prime}

 x^{\prime },y^{\prime \prime }
Derivative dots

\dot{x}, \ddot{x}

 {\dot {x}},{\ddot {x}}
Underlines, overlines, vectors

\hat a\ \bar b\ \vec c

 {\hat {a}}\ {\bar {b}}\ {\vec {c}}

\overrightarrow{a b}\ \overleftarrow{c d}\ \widehat{d e f}

 {\overrightarrow {ab}}\ {\overleftarrow {cd}}\ {\widehat {def}}

\overline{g h i}\ \underline{j k l}

 {\overline {ghi}}\ {\underline {jkl}}

\not 1\ \cancel{123}

 \not 1\ {\cancel {123}}
Arrows

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

 A{\xleftarrow {n+\mu -1}}B{\xrightarrow[{T}]{n\pm i-1}}C
Overbraces

\overbrace{ 1+2+\cdots+100 }^{\text{sum}\,=\,5050}

 \overbrace {1+2+\cdots +100} ^{{\text{sum}}\,=\,5050}
Underbraces

\underbrace{ a+b+\cdots+z }_{26\text{ terms}}

 \underbrace {a+b+\cdots +z} _{26{\text{ terms}}}
Sum

\sum_{k=1}^N k^2

 \sum _{k=1}^{N}k^{2}
Sum (force \textstyle)

\textstyle \sum_{k=1}^N k^2

 \textstyle \sum _{k=1}^{N}k^{2}
Product

\prod_{i=1}^N x_i

 \prod _{i=1}^{N}x_{i}
Product (force \textstyle)

\textstyle \prod_{i=1}^N x_i

 \textstyle \prod _{i=1}^{N}x_{i}
Coproduct

\coprod_{i=1}^N x_i

 \coprod _{i=1}^{N}x_{i}
Coproduct (force \textstyle)

\textstyle \coprod_{i=1}^N x_i

 \textstyle \coprod _{i=1}^{N}x_{i}
Limit

\lim_{n \to \infty}x_n

 \lim _{n\to \infty }x_{n}
Limit (force \textstyle)

\textstyle \lim_{n \to \infty}x_n

 \textstyle \lim _{n\to \infty }x_{n}
Integral

\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int \limits _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (alternate limits style)

\int_{1}^{3}\frac{e^3/x}{x^2}\, dx

 \int _{1}^{3}{\frac {e^{3}/x}{x^{2}}}\,dx
Integral (force \textstyle)

\textstyle \int\limits_{-N}^{N} e^x\, dx

 \textstyle \int \limits _{-N}^{N}e^{x}\,dx
Integral (force \textstyle, alternate limits style)

\textstyle \int_{-N}^{N} e^x\, dx

 \textstyle \int _{-N}^{N}e^{x}\,dx
Double integral

\iint\limits_D \, dx\,dy

 \iint \limits _{D}\,dx\,dy
Triple integral

\iiint\limits_E \, dx\,dy\,dz

 \iiint \limits _{E}\,dx\,dy\,dz
Quadruple integral

\iiiint\limits_F \, dx\,dy\,dz\,dt

 \iiiint \limits _{F}\,dx\,dy\,dz\,dt
Line or path integral

\int_C x^3\, dx + 4y^2\, dy

 \int _{C}x^{3}\,dx+4y^{2}\,dy
Closed line or path integral

\oint_C x^3\, dx + 4y^2\, dy

 \oint _{C}x^{3}\,dx+4y^{2}\,dy
Intersections

\bigcap_1^n p

 \bigcap _{1}^{n}p
Unions

\bigcup_1^k p

 \bigcup _{1}^{k}p

Fractions, matrices, multi-lines

Feature Syntax How it looks rendered
Fractions

\frac{1}{2}=0.5

 {\frac {1}{2}}=0.5
Small ("text style") fractions

\tfrac{1}{2} = 0.5

 {\tfrac {1}{2}}=0.5
Large ("display style") fractions

\dfrac{k}{k-1} = 0.5

 {\dfrac {k}{k-1}}=0.5
Mixture of large and small fractions

\dfrac{ \tfrac{1}{2}[1-(\tfrac{1}{2})^n] }{ 1-\tfrac{1}{2} } = s_n

 {\dfrac {{\tfrac {1}{2}}[1-({\tfrac {1}{2}})^{n}]}{1-{\tfrac {1}{2}}}}=s_{n}
Continued fractions (note the difference in formatting)

\cfrac{2}{ c + \cfrac{2}{ d + \cfrac{1}{2} } } = a \qquad \dfrac{2}{ c + \dfrac{2}{ d + \dfrac{1}{2} } } = a

 {\cfrac {2}{c+{\cfrac {2}{d+{\cfrac {1}{2}}}}}}=a\qquad {\dfrac {2}{c+{\dfrac {2}{d+{\dfrac {1}{2}}}}}}=a
Binomial coefficients

\binom{n}{k}

 {\binom {n}{k}}
Small ("text style") binomial coefficients

\tbinom{n}{k}

 {\tbinom {n}{k}}
Large ("display style") binomial coefficients

\dbinom{n}{k}

 {\dbinom {n}{k}}
Matrices

\begin{matrix} x & y \\ z & v \end{matrix}

 {\begin{matrix}x&y\\z&v\end{matrix}}

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

 {\begin{vmatrix}x&y\\z&v\end{vmatrix}}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

 {\begin{Vmatrix}x&y\\z&v\end{Vmatrix}}

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

 {\begin{bmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{bmatrix}}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

 {\begin{Bmatrix}x&y\\z&v\end{Bmatrix}}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

 {\begin{pmatrix}x&y\\z&v\end{pmatrix}}

\bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)

 {\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}
Arrays

\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0 \end{array}

 {\displaystyle {\begin{array}{|c|c||c|}a&b&S\\\hline 0&0&1\\0&1&1\\1&0&1\\1&1&0\end{array}}}
Cases

f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}

 f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}
System of equations

\begin{cases} 3x + 5y + z &= 1 \\ 7x - 2y + 4z &= 2 \\ -6x + 3y + 2z &= 3 \end{cases}

 {\begin{cases}3x+5y+z&=1\\7x-2y+4z&=2\\-6x+3y+2z&=3\end{cases}}
Breaking up a long expression so it wraps when necessary

<math>f(x) = \sum_{n=0}^\infty a_n x^n</math> <math>= a_0 + a_1x + a_2x^2 + \cdots</math>

 f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
Multiline equations

\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \end{align}

 {\displaystyle {\begin{aligned}f(x)&=(a+b)^{2}\\&=a^{2}+2ab+b^{2}\end{aligned}}}

\begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \end{alignat}

 {\displaystyle {\begin{alignedat}{2}f(x)&=(a-b)^{2}\\&=a^{2}-2ab+b^{2}\end{alignedat}}}
Multiline equations with alignment specified (left, center, right)

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcl}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

 {\begin{array}{lcr}z&=&a\\f(x,y,z)&=&x+y+z\end{array}}

Parenthesizing big expressions, brackets, bars

Feature Syntax How it looks rendered
Bad

( \frac{1}{2} )

 ({\frac {1}{2}})
Good

\left ( \frac{1}{2} \right )

 \left({\frac {1}{2}}\right)

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses

\left ( \frac{a}{b} \right )

 \left({\frac {a}{b}}\right)
Brackets

\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

 \left[{\frac {a}{b}}\right]\quad \left\lbrack {\frac {a}{b}}\right\rbrack
Braces (note the backslash before the braces in the code)

\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

 \left\{{\frac {a}{b}}\right\}\quad \left\lbrace {\frac {a}{b}}\right\rbrace
Angle brackets

\left \langle \frac{a}{b} \right \rangle

 \left\langle {\frac {a}{b}}\right\rangle
Bars and double bars (note: "bars" provide the absolute value function)

\left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|

 \left|{\frac {a}{b}}\right\vert \left\Vert {\frac {c}{d}}\right\|
Floor and ceiling functions:

\left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil

 \left\lfloor {\frac {a}{b}}\right\rfloor \left\lceil {\frac {c}{d}}\right\rceil
Slashes and backslashes

\left / \frac{a}{b} \right \backslash

 \left/{\frac {a}{b}}\right\backslash
Up, down and up-down arrows

\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

 \left\uparrow {\frac {a}{b}}\right\downarrow \quad \left\Uparrow {\frac {a}{b}}\right\Downarrow \quad \left\updownarrow {\frac {a}{b}}\right\Updownarrow
Delimiters can be mixed, as long as \left and \right are both used

\left [ 0,1 \right ) \left \langle \psi \right |

 \left[0,1\right)
\left\langle \psi \right|
Use \left. or \right. if you don't want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

 \left.{\frac {A}{B}}\right\}\to X
Size of the delimiters

\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]

 {\big (}{\Big (}{\bigg (}{\Bigg (}\dots {\Bigg ]}{\bigg ]}{\Big ]}{\big ]}

\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle

\Big\rangle \big\rangle

 {\big \{}{\Big \{}{\bigg \{}{\Bigg \{}\dots {\Bigg \rangle }{\bigg \rangle }{\Big \rangle }{\big \rangle }

\big| \Big| \bigg| \Bigg| \dots \Bigg\| \bigg\| \Big\| \big\|

 {\big |}{\Big |}{\bigg |}{\Bigg |}\dots {\Bigg \|}{\bigg \|}{\Big \|}{\big \|}

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \dots \Bigg\rceil

\bigg\rceil \Big\rceil \big\rceil

 {\big \lfloor }{\Big \lfloor }{\bigg \lfloor }{\Bigg \lfloor }\dots {\Bigg \rceil }{\bigg \rceil }{\Big \rceil }{\big \rceil }

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow \dots \Bigg\Downarrow

\bigg\Downarrow \Big\Downarrow \big\Downarrow

 {\big \uparrow }{\Big \uparrow }{\bigg \uparrow }{\Bigg \uparrow }\dots {\Bigg \Downarrow }{\bigg \Downarrow }{\Big \Downarrow }{\big \Downarrow }

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow \dots

\Bigg\Updownarrow \bigg\Updownarrow \Big\Updownarrow \big\Updownarrow

 {\big \updownarrow }{\Big \updownarrow }{\bigg \updownarrow }{\Bigg \updownarrow }\dots {\Bigg \Updownarrow }{\bigg \Updownarrow }{\Big \Updownarrow }{\big \Updownarrow }

\big / \Big / \bigg / \Bigg / \dots \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

 {\big /}{\Big /}{\bigg /}{\Bigg /}\dots {\Bigg \backslash }{\bigg \backslash }{\Big \backslash }{\big \backslash }

Alphabets

Greek alphabet
Boldface (greek)

\Alpha \Beta \Gamma \Delta \Epsilon \Zeta

 \mathrm {A} \mathrm {B} \Gamma \Delta \mathrm {E} \mathrm {Z} \,

\Eta \Theta \Iota \Kappa \Lambda \Mu

 \mathrm {H} \Theta \mathrm {I} \mathrm {K} \Lambda \mathrm {M} \,

\Nu \Xi \Omicron \Pi \Rho \Sigma \Tau

 \mathrm {N} \Xi \mathrm {O} \Pi \mathrm {P} \Sigma \mathrm {T} \,

\Upsilon \Phi \Chi \Psi \Omega

 \Upsilon \Phi \mathrm {X} \Psi \Omega \,

\alpha \beta \gamma \delta \epsilon \zeta

 \alpha \beta \gamma \delta \epsilon \zeta \,

\eta \theta \iota \kappa \lambda \mu

 \eta \theta \iota \kappa \lambda \mu \,

\nu \xi \omicron \pi \rho \sigma \tau

 {\displaystyle \nu \xi \mathrm {o} \pi \rho \sigma \tau \,}

\upsilon \phi \chi \psi \omega

 \upsilon \phi \chi \psi \omega \,

\varepsilon \digamma \vartheta \varkappa

 \varepsilon \digamma \vartheta \varkappa \,

\varpi \varrho \varsigma \varphi

 \varpi \varrho \varsigma \varphi \,

\boldsymbol{\Alpha} \boldsymbol{\Beta} \boldsymbol{\Gamma} \boldsymbol{\Delta} \boldsymbol{\Epsilon} \boldsymbol{\Zeta}

 {\boldsymbol {\mathrm {A} }}{\boldsymbol {\mathrm {B} }}{\boldsymbol {\Gamma }}{\boldsymbol {\Delta }}{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {Z} }}\,

\boldsymbol{\Eta} \boldsymbol{\Theta} \boldsymbol{\Iota} \boldsymbol{\Kappa} \boldsymbol{\Lambda}

\boldsymbol{\Mu}

 {\boldsymbol {\mathrm {H} }}{\boldsymbol {\Theta }}{\boldsymbol {\mathrm {I} }}{\boldsymbol {\mathrm {K} }}{\boldsymbol {\Lambda }}{\boldsymbol {\mathrm {M} }}\,

\boldsymbol{\Nu} \boldsymbol{\Xi} \boldsymbol{\Pi} \boldsymbol{\Rho} \boldsymbol{\Sigma}

\boldsymbol{\Tau}

 {\boldsymbol {\mathrm {N} }}{\boldsymbol {\Xi }}{\boldsymbol {\Pi }}{\boldsymbol {\mathrm {P} }}{\boldsymbol {\Sigma }}{\boldsymbol {\mathrm {T} }}\,

\boldsymbol{\Upsilon} \boldsymbol{\Phi} \boldsymbol{\Chi} \boldsymbol{\Psi} \boldsymbol{\Omega}

 {\boldsymbol {\Upsilon }}{\boldsymbol {\Phi }}{\boldsymbol {\mathrm {X} }}{\boldsymbol {\Psi }}{\boldsymbol {\Omega }}\,

\boldsymbol{\alpha} \boldsymbol{\beta} \boldsymbol{\gamma} \boldsymbol{\delta} \boldsymbol{\epsilon}

\boldsymbol{\zeta}

 {\boldsymbol {\alpha }}{\boldsymbol {\beta }}{\boldsymbol {\gamma }}{\boldsymbol {\delta }}{\boldsymbol {\epsilon }}{\boldsymbol {\zeta }}\,

\boldsymbol{\eta} \boldsymbol{\theta} \boldsymbol{\iota} \boldsymbol{\kappa} \boldsymbol{\lambda}

\boldsymbol{\mu}

 {\boldsymbol {\eta }}{\boldsymbol {\theta }}{\boldsymbol {\iota }}{\boldsymbol {\kappa }}{\boldsymbol {\lambda }}{\boldsymbol {\mu }}\,

\boldsymbol{\nu} \boldsymbol{\xi} \boldsymbol{\pi} \boldsymbol{\rho} \boldsymbol{\sigma}

\boldsymbol{\tau}

 {\boldsymbol {\nu }}{\boldsymbol {\xi }}{\boldsymbol {\pi }}{\boldsymbol {\rho }}{\boldsymbol {\sigma }}{\boldsymbol {\tau }}\,

\boldsymbol{\upsilon} \boldsymbol{\phi} \boldsymbol{\chi} \boldsymbol{\psi} \boldsymbol{\omega}

 {\boldsymbol {\upsilon }}{\boldsymbol {\phi }}{\boldsymbol {\chi }}{\boldsymbol {\psi }}{\boldsymbol {\omega }}\,

\boldsymbol{\varepsilon} \boldsymbol{\digamma} \boldsymbol{\vartheta} \boldsymbol{\varkappa}

 {\boldsymbol {\varepsilon }}{\boldsymbol {\digamma }}{\boldsymbol {\vartheta }}{\boldsymbol {\varkappa }}\,

\boldsymbol{\varpi} \boldsymbol{\varrho} \boldsymbol{\varsigma} \boldsymbol{\varphi}

 {\boldsymbol {\varpi }}{\boldsymbol {\varrho }}{\boldsymbol {\varsigma }}{\boldsymbol {\varphi }}\,

References:


]]>[/allow-dzen] [/yandexrss][shortrss] Two Youtube channels of mathematics https://farid.partonia.ir/index.php?newsid=9 https://farid.partonia.ir/index.php?newsid=9

The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

 [allow-turbo]The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]>[/allow-turbo] Mathematics FariD Sat, 07 Aug 2021 23:20:30 +0430 [/shortrss] [fullrss] Two Youtube channels of mathematics https://farid.partonia.ir/index.php?newsid=9 https://farid.partonia.ir/index.php?newsid=9 FariD Sat, 07 Aug 2021 23:20:30 +0430 The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]> [allow-turbo]The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]>[/allow-turbo] [allow-dzen]The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]>[/allow-dzen] [/fullrss] [yandexrss] Two Youtube channels of mathematics https://farid.partonia.ir/index.php?newsid=9

The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

 Mathematics Sat, 07 Aug 2021 23:20:30 +0430

The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

 [allow-turbo]The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]>[/allow-turbo] [allow-dzen]The below Youtube channels are highly recommended for bachelors since they will help to understand and visualize the semantics of math.

Dr. Trefor Bazett

Wild Egg Maths

]]>[/allow-dzen] [/yandexrss][shortrss] Applied Numerical Analysis - F.Gerald O. Wheatley https://farid.partonia.ir/index.php?newsid=8 https://farid.partonia.ir/index.php?newsid=8

This book is profoundly helpful in the case that you want to commence realizing how to apply mathematics in real-world problems; moreover, it is a general book since it only includes the algorithm pseudo-code.

By and large, this is old but gold, do not miss reading this book if you are passionate about mathematics and programming.

The list of contents and download links are available on the full article.

 [allow-turbo]Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

]]>[/allow-turbo] Numerical Methods FariD Sat, 07 Aug 2021 21:17:27 +0430 [/shortrss] [fullrss] Applied Numerical Analysis - F.Gerald O. Wheatley https://farid.partonia.ir/index.php?newsid=8 https://farid.partonia.ir/index.php?newsid=8 FariD Sat, 07 Aug 2021 21:17:27 +0430 This book is profoundly helpful in the case that you want to commence realizing how to apply mathematics in real-world problems; moreover, it is a general book since it only includes the algorithm pseudo-code.

By and large, this is old but gold, do not miss reading this book if you are passionate about mathematics and programming.

The list of contents and download links are available on the full article.

]]> [allow-turbo]Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

]]>[/allow-turbo] [allow-dzen]Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

]]>[/allow-dzen] [/fullrss] [yandexrss] Applied Numerical Analysis - F.Gerald O. Wheatley https://farid.partonia.ir/index.php?newsid=8

This book is profoundly helpful in the case that you want to commence realizing how to apply mathematics in real-world problems; moreover, it is a general book since it only includes the algorithm pseudo-code.

By and large, this is old but gold, do not miss reading this book if you are passionate about mathematics and programming.

The list of contents and download links are available on the full article.

 Numerical Methods Sat, 07 Aug 2021 21:17:27 +0430

Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

 [allow-turbo]Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

]]>[/allow-turbo] [allow-dzen]Table of Contents:

Analysis Versus Numerical Analysis

  • Computers and Numerical Analysis
  • An Illustrative Example
  • Kinds of Errors in Numerical Procedures
  • Interval Arithmetic
  • Parallel and Distributed Computing
  • Measuring the Efficiency of Numerical Procedures
  • Exercises
  • Applied Problems and Projects

Solving Nonlinear Equations

  • Interval Halving (Bisection)
  • Linear Interpolation Methods
  • Newton's Method
  • Muller's Method
  • Fixed-Point Iteration: x = g(x) Method
  • Multiple Roots
  • Nonlinear Systems
  • Exercises
  • Applied Problems and Projects

Solving Sets of Equations

  • Matrices and Vectors
  • Elimination Methods
  • The Inverse of a Matrix and Matrix Pathology
  • Ill-Conditioned Systems
  • Iterative Methods
  • Parallel Processing
  • Exercises
  • Applied Problems and Projects

Interpolation and Curve Fitting

  • Interpolating Polynomials
  • Divided Differences
  • Spline Curves
  • Bezier Curves and B-Splines Curves
  • Interpolating on a Surface
  • Least-Squares Approximations
  • Exercises
  • Applied Problems and Projects

Approximation of Functions

  • Chebyshev Polynomials and Chebyshev Series
  • Rational Function Approximations
  • Fourier Series
  • Exercises
  • Applied Problems and Projects

Numerical Differentiation and Integration

  • Differentiation with a Computer
  • Numerical Integration-The Trapezoidal Rule
  • Simpson's Rules
  • An Application of Numerical Integration-Fourier Series and Fourier Transforms
  • Adaptive Integration
  • Gaussian Quadrature
  • Multiple Integrals
  • Applications of Cubic Splines
  • Exercises
  • Applied Problems and Projects

Numerical Solution of Ordinary

  • Differential Equations
  • The Taylor-Series Method
  • The Euler Method and Its Modifications
  • Runge-Kutta Methods
  • Multistep Methods
  • Higher-Order Equations and Systems
  • Stiff Equations
  • Boundary-Value Problems
  • Characteristic-Value Problems
  • Exercises
  • Applied Problems and Projects

Optimization

  • Finding the Minimum of y = f(x)
  • Minimizing a Function of Several Variables
  • Linear Programming
  • Nonlinear Programming
  • Other Optimizations
  • Exercises
  • Applied Problems and Projects

Partial-Differential Equations

  • Elliptic Equations
  • Parabolic Equations
  • Hyperbolic Equations
  • Exercises
  • Applied Problems and Projects

Finite Element Analysis

  • Mathematical Background
  • Finite Elements for Ordinary-Differential Equations
  • Finite Elements for Partial-Differential Equations
  • Exercises
  • Applied Problems and Projects

Appendixes

  • A Some Basic Information from Calculus
  • B Software Resources
  • Answers to Selected Exercises
  • References

Click to Download 7th Edition with corresponding manual

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