# Using TeX notation

When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

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# Several examples of TeX equations

## The Lorenz Equations

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

## The Cauchy-Schwarz Inequality

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

## A Cross Product Formula

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

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

\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]

## An Identity of Ramanujan

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

## A Rogers-Ramanujan Identity

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

## Maxwell's Equations

\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-line Mathematics

Finally, while display equations look good for a page of samples, the
ability to mix math and text in a paragraph is also important.  This
expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an inline equation.  As
you see, MathJax equations can be used this way as well, without unduly
disturbing the spacing between lines.

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# Misc

- $E = mc^2$

- \( A_i = B_i + C_i \sum_{k=0}^{i} D_k E^k \)

- \begin{eqnarray}
  A_i &=& B_i + C_i \sum_{k=0}^{i} D_k E^k \\
  F_i &=& \int_{-\infty}^{x_i} f(x) dx
\end{eqnarray}

- $\frac{w_x}{\sum_z x_z}$
- $\frac{w}{\sum_{z} x_z}$
- $x_\gamma = x_i$
- $x_i = x_\gamma$

Cost function of logistic regression (revision):

$$J(\theta) = - \frac{1}{m} \sum_{i=1}^m [ y^{(i)}\ \log (h_\theta (x^{(i)})) + (1 - y^{(i)})\ \log (1 - h_\theta(x^{(i)}))] + \frac{\lambda}{2m}\sum_{j=1}^n \theta_j^2$$

For Neural Networks, it is:

$$
J(\Theta) = - \frac{1}{m} \sum_{i=1}^m \sum_{k=1}^K \left[y^{(i)}_k \log ((h_\Theta (x^{(i)}))_k) + (1 - y^{(i)}_k)\log (1 - (h_\Theta(x^{(i)}))_k)\right] + \frac{\lambda}{2m}\sum_{l=1}^{L-1} \sum_{p=1}^{s_l} \sum_{n=1}^{s_{l+1}} ( \Theta_{n,p}^{(l)})^2
$$