Basic Latex Usage

1. Simple Inline Equations:
   markdown

   Here are some inline equations:
   - Area of a circle: $A = \pi r^2$
   - Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
   - Einstein's famous equation: $E = mc^2$

   Output:Here are some inline equations:

   • Area of a circle: $A = \pi r^2$
   • Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
   • Einstein's famous equation: $E = mc^2$
2. Basic Block Equations:
   markdown

   The Gaussian integral:
   $$
   \int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}
   $$

   Output:The Gaussian integral:$$\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}$$

Advanced Mathematical Expressions

1. Matrix Operations:
   markdown

   $$
   \begin{pmatrix}
       a & b \\
       c & d
   \end{pmatrix}
   \begin{pmatrix}
       x \\
       y
   \end{pmatrix} =
   \begin{pmatrix}
       ax + by \\
       cx + dy
   \end{pmatrix}
   $$

   Output:$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$$
2. Multi-line Equations with Alignment:
   markdown

   $$
   \begin{align}
       (x + y)^3 &= (x + y)(x + y)^2 \\
       &= (x + y)(x^2 + 2xy + y^2) \\
       &= x^3 + 3x^2y + 3xy^2 + y^3
   \end{align}
   $$

   Output:$$\begin{align} (x + y)^3 &= (x + y)(x + y)^2 \\ &= (x + y)(x^2 + 2xy + y^2) \\ &= x^3 + 3x^2y + 3xy^2 + y^3 \end{align}$$

Other examples

$$
    A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
$$

$$
    \forall x \in \mathbb{R}, \exists y : y > x
$$

$$
    P(A|B) = \frac{P(B|A)P(A)}{P(B)}
$$

$$
    \frac{d}{dx}\left[\int_a^x f(t)dt\right] = f(x)
$$

$$
    \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}, |x| < 1
$$

$$
    \lim_{x \to 0} \frac{\sin x}{x} = 1
$$

The most beautiful equation in mathematics:
$$
    e^{i\pi} + 1 = 0
$$

$$
    \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \quad \text{(Riemann zeta function)}
$$

$$
    \Gamma(z) = \int_0^{\infty} t^{z-1}e^{-t}dt \quad \text{(Gamma function)}
$$

$$
    \vartheta(z) = \sum_{n=-\infty}^{\infty} e^{-\pi n^2 z} \quad \text{(Theta function)}
$$