Using LaTeX to Typeset Equations

LaTeX is a markup and programming language for typesetting mathematical languages. It is used extensively in Physics, Math, Statistics and other technical fields for writing papers and other documents with equations.

You can use LaTeX syntax in the Markdown cells of you Jupyer notebooks. This enables you to create notebooks with have equations in addition to code and narrative text. The Jupyter Notebook renders LaTeX using the excellent MathJax library.

Learning LaTeX

If you don't already know LaTeX syntax, a good starting place is the Short Math Guide to LaTeX by the American Mathematical Society.

The rest of this document show examples of LaTeX in the notebook.

The Lorenz equations

Source

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

Display

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

The Cauchy-Schwarz inequality

Source

\begin{equation*}
\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)
\end{equation*}

Display

\begin{equation*} \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) \end{equation*}

A cross product formula

Source

\begin{equation*}
\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}  
\end{equation*}

Display

\begin{equation*} \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} \end{equation*}

The probability of getting (k) heads when flipping (n) coins is

Source

\begin{equation*}
P(E)   = {n \choose k} p^k (1-p)^{ n-k} 
\end{equation*}

Display

\begin{equation*} P(E) = {n \choose k} p^k (1-p)^{ n-k} \end{equation*}

An identity of Ramanujan

Source

\begin{equation*}
\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+\ldots} } } } 
\end{equation*}

Display

\begin{equation*} \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+\ldots} } } } \end{equation*}

A Rogers-Ramanujan identity

Source

\begin{equation*}
1 +  \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \text{for $|q|<1$}. 
\end{equation*}

Display

\begin{equation*} 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for $|q|<1$}. \end{equation*}

Maxwell's equations

Source

\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}

Display

\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}

Equation Numbering and References

Equation numbering and referencing will be available in a future version of the Jupyter Notebook.

Inline Typesetting (Mixing Markdown and TeX)

While display equations look good for a page of samples, the ability to mix math and formatted text in a paragraph is also important.

Source

This expression $\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a **[Markdown-formatted](http://daringfireball.net/projects/markdown/)** sentence.

Display

This expression $\sqrt{3x-1}+(1+x)^2$ is an example of a TeX inline equation in a Markdown-formatted sentence.

Other syntax

You will notice in other places on the web that $$ are needed explicitly to begin and end MathJax typesetting. This is not required if you will be using TeX environments, but the IPython notebook will accept this syntax on legacy notebooks.

Source

$$
\begin{array}{c}
y_1 \\\
y_2 \mathtt{t}_i \\\
z_{3,4}
\end{array}
$$

$$
\begin{array}{c}
y_1 \cr
y_2 \mathtt{t}_i \cr
y_{3}
\end{array}
$$

$$\begin{eqnarray} 
x' &=& &x \sin\phi &+& z \cos\phi \\
z' &=& - &x \cos\phi &+& z \sin\phi \\
\end{eqnarray}$$

$$
x=4
$$

Display

$$ \begin{array}{c} y_1 \\\ y_2 \mathtt{t}_i \\\ z_{3,4} \end{array} $$$$ \begin{array}{c} y_1 \cr y_2 \mathtt{t}_i \cr y_{3} \end{array} $$$$\begin{eqnarray} x' &=& &x \sin\phi &+& z \cos\phi \\ z' &=& - &x \cos\phi &+& z \sin\phi \\ \end{eqnarray}$$$$ x=4 $$