# Math

## Unicode Characters vs LaTeX Commands

**Use UTF-8 unicode characters in narrative text, but LaTeX commands in math environments:**

### In Narrative Text
Use UTF-8 unicode characters (α, β, γ, etc.) in all narrative text **except** when inside LaTeX environments:

```md
The parameter α controls the utility function, and β represents the discount factor.
In economic models, we often see parameters like σ, θ, and ρ.
```

### In LaTeX Math Environments  
Use LaTeX commands (`\alpha`, `\beta`, etc.) inside LaTeX environments such as `$`, `$$`, and `:math:`:

```md
The utility function is $u(c) = \frac{c^{1-\alpha}}{1-\alpha}$ where $\alpha > 0$.

$$
\max \mathbb{E} \sum_{t=0}^{\infty} \beta^t u(c_t)
$$

The discount factor is {math}`\beta \in (0,1)`.
```

### In Code
Use UTF-8 unicode characters in variable names and code (see [Code Style Guide](code.md)):

```python
def utility_function(c, α=0.5, β=0.95):
    return (c**(1-α) - 1) / (1-α) * β
```

```{note}
This approach ensures proper LaTeX rendering in math environments while maintaining readability in narrative text and code.
```

## Notation

```{margin}
Example notation can be found [here](https://python.quantecon.org/lqcontrol.html)
```

Use `\top`(e.g., $A^\top$) to represent the transpose of a vector or matrix.

Use `\mathbb{1}`($\mathbb{1}$)  to represent a vector or matrix of ones and explain it in the lecture (e.g., "Let $\mathbb{1}$ be an $n \times 1$ vector of ones...").

Matrices always use square brackets and `\begin{bmatrix} ... \end{bmatrix}` should be used.

Do **not** use `bold face` for either matrices or vectors

Sequences use curly brackets, such as `\{ x_t \}_{t=0}^{\infty}`

### Probability, expectation, and variance

Use `\mathbb{P}` for probability, with braces `\{...\}` for events and parentheses `(...)` for sets:

```md
$\mathbb{P}\{X = 5\}$, $\mathbb{P}\{X \leq x\}$, $\mathbb{P}(B)$
```

Use `\mathbb{E}` for expectation and `\mathbb{V}` for variance:

```md
$\mathbb{E}[X]$, $\mathbb{V}[X]$
```

### Distribution names

Use plain letters for named distributions — not `\mathcal` or `\mathbb`. For multi-letter distribution names use `\mathrm`:

```md
$X \sim N(\mu, \sigma^2)$, $\theta \sim \mathrm{Beta}(\alpha, \beta)$
```

### Density and mass functions

Use lowercase letters (e.g., $f$, $g$, $p$) for density and mass functions, uppercase for CDFs:

```md
$f(x)$ for the density, $F(x)$ for the CDF
```

### Multiplication in equations

Use `\cdot` or juxtaposition for multiplication in LaTeX — never use `*`.

### IID

Write "IID" in text (not "i.i.d." or "iid").

The use of align environments can be done using the `\begin{aligned} ... \end{aligned}`
when inside a math environment such as `$$`

:::{tip}
**PDF:**

Using `aligned` when inside a math environment (such as `$$`) is important
for building `pdf` output

```md
$$
\begin{aligned}

\end{aligned}
$$
```

This is because the `$$` is already a math environment and causes the LaTeX build
to fail when using `align` due to the issue of multiple math environments. 

The LaTeX `aligned` environment is purpose built for using within `math` environments.
:::

It is also important to **not** use `\tag` in equations for manual equation numbering and instead
make use of the in-built equation numbering such as

```md
$$
x_y = 2
$$ (label)
```

which can be referenced using the role

```md
and here is the reference to {eq}`label`
```