# Math

Render LaTeX math in pages using [MathJax](https://www.mathjax.org/).

## Configuration

Configure Goldmark to bypass math delimiters:

```yaml {title="hugo.yaml"}
markup:
  goldmark:
    extensions:
      passthrough:
        enable: true
        delimiters:
          block:
            - ['\[', '\]']
            - ['$$', '$$']
          inline:
            - ['\(', '\)']
```

Passthrough skips Goldmark processing for matched content. Then enable math per page:

```yaml {title="index.md"}
---
title: "Example Page"
params:
  mathEnabled: true
---
```

Set `params.mathEnabled` in `hugo.yaml` for global enablement.

## Usage

Place inline formulas inside `\( ... \)`, and block formulas inside `$$` blocks.

> [!TIP]
> Goldmark passthrough forbids blank lines. The `math` shortcode allows them and works for complex equations.
>
> The `math` shortcode does not require `mathEnabled` set in front matter.

## Examples

Example of an inline equation:

```tex
The distance is \(f(a,b) = \sqrt{(a^2+b^2)}\).
```

The distance is \(f(a,b) = \sqrt{(a^2+b^2)}\).

Example of an block equation:

```tex
$$
\begin{align}
&h_{\text{LOS}} &&= e^{(j2\pi\mathcal{N}(1,1))} \\
&h_{\text{NLOS}} &&= \mathcal{CN}(0, \sigma^2) \\
&h &&= h_{\text{LOS}}\sqrt{\dfrac{K}{K + 1}} + h_{\text{NLOS}}\sqrt{\frac{1}{K + 1}}
\end{align}
$$
```

$$
\begin{align}
&h_{\text{LOS}} &&= e^{(j2\pi\mathcal{N}(1,1))} \\
&h_{\text{NLOS}} &&= \mathcal{CN}(0, \sigma^2) \\
&h &&= h_{\text{LOS}}\sqrt{\dfrac{K}{K + 1}} + h_{\text{NLOS}}\sqrt{\frac{1}{K + 1}}
\end{align}
$$

Example of an complex equation using `math` shortcode:

```tex
{{</* math >}}
$$
\begin{array}{l}

  \mathbf{\text{Algorithm 1: Block Orthogonal Matching Pursuit (BOMP)}} \\

  \hline \\

  \textbf{Input:} \text{ Measurement } \mathbf{y} \in \mathbb{C}^{M}, \text{ Dictionary } \mathbf{\Phi} = [\mathbf{\Phi}_1, \dots, \mathbf{\Phi}_L] \in \mathbb{C}^{M \times N}, \text{ Sparsity } K. \\
  \textbf{Output:} \text{ Estimate } \hat{\mathbf{x}} \in \mathbb{C}^{N}. \\

  \hline \\

  \textbf{Initialization:} \\
    \quad 1. \text{ Residual: } \mathbf{r}^0 \leftarrow \mathbf{y} \\
    \quad 2. \text{ Block Support: } \mathbf{\Omega}^0 \leftarrow \emptyset \\
    \quad 3. \text{ Iteration: } k \leftarrow 1 \\
  \\
  \mathbf{\text{while }} k \le K \mathbf{\text{ do}} \\
    \quad 4. \quad \text{Block Selection:} \quad j_k \leftarrow \arg \max_{j \notin \mathbf{\Omega}^{k-1}} \left\| \mathbf{\Phi}_j^H \mathbf{r}^{k-1} \right\|_2 \\
    \quad 5. \quad \text{Support Update:} \quad \mathbf{\Omega}^k \leftarrow \mathbf{\Omega}^{k-1} \cup \{j_k\} \\
    \quad 6. \quad \text{Sub-dictionary:} \quad \mathbf{\Phi}_{\mathbf{\Omega}^k} \leftarrow [\mathbf{\Phi}_j \mid j \in \mathbf{\Omega}^k] \\
    \quad 7. \quad \text{Coefficient Solve (LSE):} \quad \mathbf{c}^k \leftarrow \mathbf{\Phi}_{\mathbf{\Omega}^k}^{\dagger} \mathbf{y} \\
    \quad 8. \quad \text{Residual Update:} \quad \mathbf{r}^k \leftarrow \mathbf{y} - \mathbf{\Phi}_{\mathbf{\Omega}^k} \mathbf{c}^k \\
    \quad 9. \quad k \leftarrow k + 1 \\
  \mathbf{\text{End while}} \\
  \\

  \textbf{Return:} \\
    \quad 10. \text{ Reconstruct } \hat{\mathbf{x}} \text{ using coefficients } \mathbf{c}^K \text{ on support } \mathbf{\Omega}^K \text{ and zeros elsewhere.} \\

  \hline

\end{array}
$$
{{< /math */>}}
```

{{< math >}}
$$
\begin{array}{l}

  \mathbf{\text{Algorithm 1: Block Orthogonal Matching Pursuit (BOMP)}} \\

  \hline \\

  \textbf{Input:} \text{ Measurement } \mathbf{y} \in \mathbb{C}^{M}, \text{ Dictionary } \mathbf{\Phi} = [\mathbf{\Phi}_1, \dots, \mathbf{\Phi}_L] \in \mathbb{C}^{M \times N}, \text{ Sparsity } K. \\
  \textbf{Output:} \text{ Estimate } \hat{\mathbf{x}} \in \mathbb{C}^{N}. \\

  \hline \\

  \textbf{Initialization:} \\
    \quad 1. \text{ Residual: } \mathbf{r}^0 \leftarrow \mathbf{y} \\
    \quad 2. \text{ Block Support: } \mathbf{\Omega}^0 \leftarrow \emptyset \\
    \quad 3. \text{ Iteration: } k \leftarrow 1 \\
  \\
  \mathbf{\text{while }} k \le K \mathbf{\text{ do}} \\
    \quad 4. \quad \text{Block Selection:} \quad j_k \leftarrow \arg \max_{j \notin \mathbf{\Omega}^{k-1}} \left\| \mathbf{\Phi}_j^H \mathbf{r}^{k-1} \right\|_2 \\
    \quad 5. \quad \text{Support Update:} \quad \mathbf{\Omega}^k \leftarrow \mathbf{\Omega}^{k-1} \cup \{j_k\} \\
    \quad 6. \quad \text{Sub-dictionary:} \quad \mathbf{\Phi}_{\mathbf{\Omega}^k} \leftarrow [\mathbf{\Phi}_j \mid j \in \mathbf{\Omega}^k] \\
    \quad 7. \quad \text{Coefficient Solve (LSE):} \quad \mathbf{c}^k \leftarrow \mathbf{\Phi}_{\mathbf{\Omega}^k}^{\dagger} \mathbf{y} \\
    \quad 8. \quad \text{Residual Update:} \quad \mathbf{r}^k \leftarrow \mathbf{y} - \mathbf{\Phi}_{\mathbf{\Omega}^k} \mathbf{c}^k \\
    \quad 9. \quad k \leftarrow k + 1 \\
  \mathbf{\text{End while}} \\
  \\

  \textbf{Return:} \\
    \quad 10. \text{ Reconstruct } \hat{\mathbf{x}} \text{ using coefficients } \mathbf{c}^K \text{ on support } \mathbf{\Omega}^K \text{ and zeros elsewhere.} \\

  \hline

\end{array}
$$
{{< /math >}}