Write formulas

You can use MkDocs Workspace to create powerful mathematical documentation websites or print documents, by exporting docs to pdf files. MkDocs Workspace renders LaTeX math equations with MathJax.

\[ T^{i_1 i_2 \dots i_p}_{j_1 j_2 \dots j_q} = T(x^{i_1},\dots,x^{i_p},e_{j_1},\dots,e_{j_q}) \]

$$
\begin{align}
    p(v_i=1|\mathbf{h}) & = \sigma\left(\sum_j w_{ij}h_j + b_i\right) \\
    p(h_j=1|\mathbf{v}) & = \sigma\left(\sum_i w_{ij}v_i + c_j\right)
\end{align}
$$

$$
    \begin{matrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{matrix}
$$

$$ \left[
\begin{array}{cc|c}
1&2&3\\
4&5&6
\end{array}
\right] 
$$

$$
\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ 
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ 
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align} 
$$

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

$$
\begin{array}{c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 & 1+10i
\end{array}
$$

$$ \left\{ \begin{array}{l}
0 = c_x-a_{x0}-d_{x0}\dfrac{(c_x-a_{x0})\cdot d_{x0}}{\|d_{x0}\|^2} + c_x-a_{x1}-d_{x1}\dfrac{(c_x-a_{x1})\cdot d_{x1}}{\|d_{x1}\|^2} \\[2ex] 
0 = c_y-a_{y0}-d_{y0}\dfrac{(c_y-a_{y0})\cdot d_{y0}}{\|d_{y0}\|^2} + c_y-a_{y1}-d_{y1}\dfrac{(c_y-a_{y1})\cdot d_{y1}}{\|d_{y1}\|^2} \end{array} \right. 
$$

$$
\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}
$$

$$ \bbox[yellow]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (1)
}
$$

$$
|x|, ||v|| \quad\longrightarrow\quad \lvert x\rvert, \lVert v\rVert
$$

$$
\begin{array}{c|rrrr}& x^3 & x^2 & x^1 &  x^0\\ & 1 & -6 & 11 & -6\\ {\color{red}1} & \downarrow & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & |\phantom{-} {\color{blue}0} \end{array}
$$

$$
z = \overbrace{
   \underbrace{x}_\text{real} + i
   \underbrace{y}_\text{imaginary}
  }^\text{complex number}
$$

Inline vs. block mode

Formulas are created using one of the notations:

• $...$ and \(...\) for inline math
• $$...$$, \[...\], and \begin{}...\end{} for block math.

Block mode should start from the new line.

Inline and block formulas render differently. 

For example, this is $\sum_{i=0}^n i^2 = \frac{(n^2+\epsilon)(2n+1)}{6\phi}$
an inline mode) 

And the block mode looks like this (don't forget empty line!)

$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$

Add Greek letters to the formula

$$
\sum_{i=0}^n \pi^2 = \Delta\frac{(n^2+\epsilon)(2n+1)}{6\phi}
$$

Groups

A group is either a single symbol, or any formula surrounded by curly braces {…}. Operators, fractions, subscripts or superscripts appy to the group or groups.

For example, you can use use ^ and _ to create superscripts and subscripts respectively.

$$
x_i^2 + \log_2 x 
$$

And because superscripts, subscripts, and other operations apply only to the next group, if you write 10^10, you will get a surprise: \(10^10\). But 10^{10} gives what you probably wanted: \(10^{10}\).

Fractions

In order to create fractions use \frac{...}{...}. I.e.

$$
\frac{73^2-1}{12^2}
$$

Combine Fractions

$$
\frac{\Gamma + \Omega}{\frac{73^2-1}{12^2}}
$$

Operators and other notations

Examples of Latex math operators and notations

Special symbols

Sums, products, integrals

Set operators

Parentheses

Ordinary symbols ()[] make parentheses and brackets, i.e. \((2+3)[4+4]\). Use escape char \{ and \} for curly braces {}.

There are different sizes of parentheses

$$
\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)
$$

Symbols like ( will make small parentheses, which are not suitable for fractions. For example, in case of (\frac{\sqrt x}{y^3}) the output is:

$$
(\frac{\sqrt x}{y^3})
$$

But if you use \left( and \right) notations instead, you will get

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

Combine fractions, operators and parentheses:

$$
\left(\frac{\left(\sqrt{\frac{73^2}{12x}}\sqrt{\frac{x|x|}{\log_x}}\right)}{\sqrt[3]{\frac xy}}\right)
$$

System of equations, formulas and functions

Use cases to create systems of equations or functions definitions by cases (piecewise functions).

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

Aligned systems of formulas using \begin{align} and \end{align}

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

In order to create system of quations use \begin{array}, \end{array} together with \left\{ and \right

$$
\left\{ 
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\ 
a_2x+b_2y+c_2z=d_2 \\ 
a_3x+b_3y+c_3z=d_3
\end{array}
\right. 
$$

Alternative way to produce system of equations

$$
\begin{cases}
a_1x+b_1y+c_1z=d_1 \\ 
a_2x+b_2y+c_2z=d_2 \\ 
a_3x+b_3y+c_3z=d_3
\end{cases}
$$

To align the ```=``` signs use ```\begin{aligned}``` and ```\end{aligned}``` together with ```\left\{``` and ```\right```

$$
\left\{
\begin{aligned} 
a_1x+b_1y+c_1z &=d_1+e_1 \\ 
a_2x+b_2y&=d_2 \\ 
a_3x+b_3y+c_3z &=d_3 
\end{aligned} 
\right. 
$$

Use ```\begin{array}{ll}``` to align everything left

$$
\left\{
\begin{array}{ll}
a_1x+b_1y+c_1z &=d_1+e_1 \\ 
a_2x+b_2y &=d_2 \\ 
a_3x+b_3y+c_3z &=d_3 
\end{array} 
\right.
$$

Use ```\\[2ex]``` to create larger vertical spaces between equations

$$
\begin{cases}
a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\[2ex] 
a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\[2ex] 
a_3x+b_3y+c_3z=\frac{p_3}{q_3}
\end{cases}
$$

Matrices

In order to create matrixes use $$\begin{matrix}…\end{matrix}$$, in between the \begin and \end, put the matrix elements. End each matrix row with \\, and separate matrix elements with &

$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$

To add brackets use ```{pmatrix}```, ```{bmatrix}```, ```{Bmatrix}```, ```{vmatrix}```.

With ```{pmatrix}```

$$
    \begin{pmatrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{pmatrix}
$$  

With ```{bmatrix}```

$$
    \begin{bmatrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{bmatrix}
$$ 

With ```{Bmatrix}```

$$
    \begin{Bmatrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{Bmatrix}
$$  

With ```{Vmatrix}```

$$
    \begin{vmatrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{vmatrix}
$$ 

With vertical line

$$ 
\left[
\begin{array}{cc|c}
1&2&3\\
4&5&6
\end{array}
\right] 
$$

With horizontal line

$$
\begin{pmatrix}
    a & b\\
    c & d\\
\hline
    1 & 0\\
    0 & 1
\end{pmatrix}
$$

Use ```\cdots``` for **⋯**, ```\ddots``` for **⋱** and ```\vdots``` for **⋮** when you want to omit some of the entries:

$$
    \begin{vmatrix}
    1 & x & \cdots & x^2 \\
    1 & y & \cdots & y^2 \\
    \vdots & \vdots & \ddots & \vdots \\
    1 & z & \cdots & z^2 \\
    \end{vmatrix}
$$