MathJax Test

这是一个 MathJax 的测试,摘自 KaTeX and MathJax Comparison Demo

Repeating fractions

\[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } } \]

The LaTeX code:

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\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }

Summation notation

\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]

The LaTeX code:

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\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)

Sum of a Series

I broke up the next two examples into separate lines so it behaves better on a mobile phone. That's why they include .

\[\displaystyle\sum_{i=1}^{k+1}i \]

The LaTeX code:

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\displaystyle\sum_{i=1}^{k+1}i 

\[\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)\]

The LaTeX code:

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\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)

\[\displaystyle= \frac{k(k+1)}{2}+k+1\]

The LaTeX code:

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\displaystyle= \frac{k(k+1)}{2}+k+1

\[\displaystyle= \frac{k(k+1)+2(k+1)}{2}\]

The LaTeX code:

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\displaystyle= \frac{k(k+1)+2(k+1)}{2}

\[\displaystyle= \frac{(k+1)(k+2)}{2}\]

The LaTeX code:

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\displaystyle= \frac{(k+1)(k+2)}{2}

\[\displaystyle= \frac{(k+1)((k+1)+1)}{2}\]

The LaTeX code:

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\displaystyle= \frac{(k+1)((k+1)+1)}{2}

Product notation

\[ \displaystyle 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \displaystyle\text{ for }\lvert q\rvert < 1. \]

The LaTeX code:

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\displaystyle
1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots
= \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\displaystyle\text{ for }\lvert q\rvert < 1.

Inline math

And here is some in-line math: \(k_{n+1} = n^2 + k_n^2 - k_{n-1}\), followed by some more text.

The LaTeX code:

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k_{n+1} = n^2 + k_n^2 - k_{n-1}

Inline math uses

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<span class="math">...</span>

rather than a div.

Greek Letters

\[\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega\]

The LaTeX code:

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\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega

\[\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi\]

The LaTeX code:

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\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi

Arrows

\[\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow\]

The LaTeX code:

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\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow 

\[\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow\]

The LaTeX code:

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\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ 
\mapsto\ \hookleftarrow

\[\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow\]

The LaTeX code:

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\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow

\[\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup\]

The LaTeX code:

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\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ 
\hookrightarrow\ \rightharpoonup

\[\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow\]

The LaTeX code:

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\rightharpoondown\ \leadsto\ \nearrow\ 
\searrow\ \swarrow\ \nwarrow

Symbols

\[\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup\]

The LaTeX code:

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\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ 
\bigtriangleup

\[\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle\] The LaTeX code:

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\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ 
\triangleright\ \angle\ \infty\ \prime\ \triangle

Calculus

\[\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx \]

The LaTeX code:

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\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx 

\[f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}\]

The LaTeX code:

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f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}

\[\oint \vec{F} \cdot d\vec{s}=0\]

The LaTeX code:

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\oint \vec{F} \cdot d\vec{s}=0

Lorenz Equations

\[ \begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned} \]

The LaTeX code:

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\begin{aligned}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{aligned}

Cross Product

This works in KaTeX, but the separation of fractions in this environment is not so good.

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix} \]

The LaTeX code:

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\mathbf{V}_1 \times \mathbf{V}_2 =  \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
\end{vmatrix}

Here's a workaround: make the fractions smaller with an extra class that targets the spans with"mfrac" class (makes no difference in the MathJax case):

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix} \]

The LaTeX code:

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\mathbf{V}_1 \times \mathbf{V}_2 =  \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
\end{vmatrix}

Accents

\[\hat{x}\ \vec{x}\ \ddot{x} \]

The LaTeX code:

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\hat{x}\ \vec{x}\ \ddot{x} 

However, these didn't work in KaTeX:

\[\overrightarrow{AB}\]

The LaTeX code:

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\overrightarrow{AB}

\[A\underset{0^{\circ}C }{\overset{100^{\circ}C}{\rightleftarrows}}B\]

The LaTeX code:

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A\underset{0^{\circ}C }{\overset{100^{\circ}C}{\rightleftarrows}}B

Stretchy brackets

\[\left(\frac{x^2}{y^3}\right)\]

The LaTeX code:

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\left(\frac{x^2}{y^3}\right)

Evaluation at limits

\[\left.\frac{x^3}{3}\right|_0^1\]

The LaTeX code:

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\left.\frac{x^3}{3}\right|_0^1

Case definitions

\[ f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases} \]

The LaTeX code:

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

Maxwell's Equations

\[ \begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned} \]

The LaTeX code:

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\begin{aligned}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}

Statistics

Definition of combination:

\[\frac{n!}{k!(n-k)!} = {^n}C_k\]

The LaTeX code:

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\frac{n!}{k!(n-k)!} = {^n}C_k

\[{n \choose k}\]

The LaTeX code:

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{n \choose k}

Fractions on fractions

\[\frac{\frac{1}{x}+\frac{1}{y}}{y-z}\]

The LaTeX code:

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\frac{\frac{1}{x}+\frac{1}{y}}{y-z}

n-th root

\[\sqrt[n]{1+x+x^2+x^3+\ldots}\]

The LaTeX code:

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\sqrt[n]{1+x+x^2+x^3+\ldots}

Matrices

\[ \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

The LaTeX code:

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\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}

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

The LaTeX code:

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

Punctuation

\[f(x) = \sqrt{1+x} \quad (x \ge -1)\]

The LaTeX code:

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f(x) = \sqrt{1+x} \quad (x \ge  -1)

\[f(x) \sim x^2 \quad (x\to\infty)\]

The LaTeX code:

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f(x) \sim x^2 \quad (x\to\infty)

Now with punctuation:

\[f(x) = \sqrt{1+x}, \quad x \ge -1\]

The LaTeX code:

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f(x) = \sqrt{1+x}, \quad x \ge -1

\[f(x) \sim x^2, \quad x\to\infty\]

The LaTeX code:

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f(x) \sim x^2, \quad x\to\infty