数学公式

Inline Math

You can write inline math expressions like $c = \pm\sqrt{a^2 + b^2}$ or $E = mc^2$.

Block Math

Pythagorean Theorem

$$c^2 = a^2 + b^2$$

Quadratic Formula

The solutions to the quadratic equation $ax^2 + bx + c = 0$ are:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Taylor Series Expansion

The Taylor expansion of a holomorphic function $f(x)$ around point $c$:

$$f(z) = \sum_{k=0}^{\infty} \frac{(z-c)^k}{2\pi i} \int_{\gamma} \frac{f(w)}{(w-c)^{k+1}} \,dw$$

Matrix Operations

Matrix multiplication example:

$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$$

Integral Calculus

The fundamental theorem of calculus:

$$\int_a^b f'(x) \,dx = f(b) - f(a)$$

Probability

The normal distribution probability density function:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$