Math: every flavour we render

A tour of what KaTeX gives us inside the renderer. Useful as a copy-paste reference for new docs.

Inline math

Euler's identity is $e^{i\pi} + 1 = 0$ . The Cauchy-Schwarz inequality is $|\langle u, v \rangle| \le |u||v|$ — the same inequality that limits how strongly correlated two embeddings can be when their norms are bounded. We rely on it implicitly every time we threshold a cosine-similarity score.

A more workaday example: the sigmoid is $\sigma(x) = \frac{1}{1 + e^{-x}}$ , and our reranker uses softmax $\text{softmax}(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}}$ to normalise the logit scores before we threshold.

Block math

The Gaussian integral, foundational for normalising kernels:

\int_{-\infty}^{\infty} e^{-x^2}, dx = \sqrt{\pi}

The HNSW search complexity, roughly:

O(\log N) \text{ expected, with } M \cdot \log N \text{ build cost}

where $N$ is the number of indexed vectors and $M$ is the max-connections parameter we set to 16 in our pgvector index.

Matrices

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Aligned equations

\begin{aligned} y &= mx + b \\ m &= \frac{y_2 - y_1}{x_2 - x_1} \\ b &= y_1 - m \cdot x_1 \end{aligned}

What we don't render (yet)

Custom macros. KaTeX supports \newcommand, but we don't wire it through the editor — adding it would require either a per-doc macro registry or a hub-wide one. Filed as "v7 maybe" if anyone asks.