本文档用于测试在 Markdown 中使用 $$ 符号渲染数学公式。

基础代数

让我们从一些基本的代数表达式开始。

二次方程的求根公式为: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

一个简单的线性方程: $y = mx + c$

二项式平方的展开: $(a+b)^2 = a^2 + 2ab + b^2$

微积分

Here are some common expressions from calculus.

The limit definition of a derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

A definite integral: $\int_{a}^{b} f(x) dx$

The Taylor series expansion of $e^x$ around $x=0$: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

三角函数

Some basic trigonometric identities.

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$

Angle addition formula for sine: $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$

Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$

统计和概率

Formulas commonly used in statistics and probability.

The formula for the mean ($\mu$) of a set of $n$ numbers $x_1, x_2, \ldots, x_n$: $\mu = \frac{1}{n} \sum_{i=1}^{n} x_i$

The probability density function of a normal distribution: $f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Bayes’ theorem: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

线性代数

Examples from linear algebra.

A 2x2 matrix: $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

The determinant of a 2x2 matrix: $\det(A) = ad - bc$

Matrix multiplication of two matrices A and B: $C = AB$

物理学

A couple of well-known physics equations.

Einstein’s mass-energy equivalence: $E = mc^2$

Newton’s second law of motion: $F = ma$

This should provide a good test of how various mathematical formulas are rendered.

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