inline math
use single dollar signs for inline math:
| rendered | code |
|---|---|
| $E = mc^2$ | $E = mc^2$ |
| $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ | $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ |
| $e^{i\pi} + 1 = 0$ | $e^{i\pi} + 1 = 0$ |
block equations
use double dollar signs for centered blocks:
$$
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
$$
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
$$
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
$$
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$
fractions and roots
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
$$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$
\sqrt[3]{x^3 + y^3} \neq x + y
$$\sqrt[3]{x^3 + y^3} \neq x + y$$
greek letters
| rendered | code | rendered | code |
|---|---|---|---|
| $\alpha$ | \alpha |
$\Gamma$ | \Gamma |
| $\beta$ | \beta |
$\Delta$ | \Delta |
| $\gamma$ | \gamma |
$\Theta$ | \Theta |
| $\delta$ | \delta |
$\Lambda$ | \Lambda |
| $\epsilon$ | \epsilon |
$\Pi$ | \Pi |
| $\theta$ | \theta |
$\Sigma$ | \Sigma |
| $\lambda$ | \lambda |
$\Phi$ | \Phi |
| $\mu$ | \mu |
$\Omega$ | \Omega |
| $\pi$ | \pi |
||
| $\sigma$ | \sigma |
||
| $\phi$ | \phi |
||
| $\omega$ | \omega |
matrices
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
= \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}
$$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$$
\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
$$\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$
aligned equations
\begin{aligned}
(x + y)^2 &= (x + y)(x + y) \\
&= x^2 + xy + yx + y^2 \\
&= x^2 + 2xy + y^2
\end{aligned}
$$\begin{aligned} (x + y)^2 &= (x + y)(x + y) \\ &= x^2 + xy + yx + y^2 \\ &= x^2 + 2xy + y^2 \end{aligned}$$
calculus
\frac{d}{dx} \int_a^x f(t) \, dt = f(x)
$$\frac{d}{dx} \int_a^x f(t) , dt = f(x)$$
\lim_{x \to 0} \frac{\sin x}{x} = 1
$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}
$$\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k}$$
logic and sets
\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : x + y = 0
$$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : x + y = 0$$
A \cup B = \{x : x \in A \lor x \in B\}
$$A \cup B = {x : x \in A \lor x \in B}$$
A \cap B \subseteq A \cup B
$$A \cap B \subseteq A \cup B$$
common symbols
| rendered | code | rendered | code |
|---|---|---|---|
| $\leq$ | \leq |
$\partial$ | \partial |
| $\geq$ | \geq |
$\sum$ | \sum |
| $\neq$ | \neq |
$\prod$ | \prod |
| $\approx$ | \approx |
$\int$ | \int |
| $\infty$ | \infty |
$\rightarrow$ | \rightarrow |
notable formulas
gaussian distribution:
$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$
bayes’ theorem:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
shannon entropy:
$$H(X) = -\sum_{i=1}^{n} p(x_i) \log_2 p(x_i)$$
fourier transform:
$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} , dx$$
euler’s identity:
$$e^{i\pi} + 1 = 0$$