Sep 03·1 min· PV

Lab: Ratex Preview

本页面用于验证Ratex渲染效果

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

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

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

e^{i θ} = \cos θ + i \sin θ

When $θ = π$ , we get

e^{i π} = \cos π + i \sin π = - 1

and therefore

e^{i π} + 1 = 0

Here is a more complicated derivative expression:

\frac{\partial^{r}}{\partial ω^{r}} (\frac{y^{ω}}{ω}) = (\frac{y^{ω}}{ω}) {(\log y)^{r} + \sum_{i = 1}^{r} \frac{(- 1)^{i} r (r - 1) \dots (r - i + 1) (\log y)^{r - i}}{ω^{i}}}

We can also display matrices, cases, and integrals:

A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

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

\int_{0}^{1} x^{2} d x = \frac{1}{3}

Lab: Ratex Preview

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