---
title: "lab: Ratex Preview"
tags:
  - Lab
date: 2023-09-04
excerpt: 本页面用于验证Ratex渲染效果
updated: 2026-05-08 22:10:51
---

Euler’s identity $e^{i\pi}+1=0$ is often considered one of the most beautiful formulas in mathematics.

It connects the constants $e$, $i$, $\pi$, $1$, and $0$ in a single equation, and can be interpreted geometrically on the complex plane $\mathbb{R}^2$.

For example, the inline fraction $\frac{a}{b}$, the derivative $\frac{dy}{dx}$, and the limit $\lim_{x \to 0}\frac{\sin x}{x}=1$ can all be written directly inside a sentence.

$$
e^{i\theta}=\cos\theta+i\sin\theta
$$

When $\theta=\pi$, we get

$$
e^{i\pi}=\cos\pi+i\sin\pi=-1
$$

and therefore

$$
e^{i\pi}+1=0
$$

Here is a more complicated derivative expression:

$$
\frac{\partial^r}{\partial \omega^r}
\left(
  \frac{y^\omega}{\omega}
\right)
=
\left(
  \frac{y^\omega}{\omega}
\right)
\left\{
  (\log y)^r
  +
  \sum_{i=1}^{r}
  \frac{
    (-1)^i r(r-1)\cdots(r-i+1)(\log y)^{r-i}
  }{
    \omega^i
  }
\right\}
$$

We can also display matrices, cases, and integrals:

$$
A=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
$$

$$
f(x)=
\begin{cases}
x^2, & x \ge 0 \\
-x, & x < 0
\end{cases}
$$

$$
\int_0^1 x^2\,dx=\frac{1}{3}
$$