Test#002

$\KaTeX$ Test Suite

1. Inline & Display Math

• Inline: $E = mc^2$, $a^2 + b^2 = c^2$, $f(x) = \sin(x)$
• Display:

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ $$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$$

2. Greek Letters (Complete 24 Letters & Variants)

• Lowercase:

$$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \omicron, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$$

• Uppercase:

$$A, B, \Gamma, \Delta, E, Z, H, \Theta, I, K, \Lambda, M, N, \Xi, O, \Pi, P, \Sigma, T, \Upsilon, \Phi, X, \Psi, \Omega$$

• Variants:

$$\varepsilon, \vartheta, \varkappa, \varpi, \varrho, \varsigma, \varphi, \varGamma, \varDelta, \varTheta, \varLambda, \varXi, \varPi, \varSigma, \varUpsilon, \varPhi, \varPsi, \varOmega$$

3. Mathematical Symbols & Logic

• Operators & Relations: $\pm, \mp, \times, \div, \cdot, \neq, \le, \ge, \approx, \equiv, \propto, \infty, \partial, \nabla$
• Sets & Logic: $\in, \notin, \subset, \subseteq, \cup, \cap, \emptyset, \forall, \exists, \implies, \iff, \to, \mapsto, \mathbb{R}, \mathbb{C}, \mathbb{Z}, \mathbb{N}$

4. Structures (Matrices, Cases, Alignment)

Matrices & Delimiters

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \begin{vmatrix} x & y \\ z & w \end{vmatrix}, \quad \left( \frac{\sum_{i=1}^n x_i}{\prod_{j=1}^m y_j} \right)$$

Cases & Alignment

$$f(x) = \begin{cases} -x & (x < 0) \\ 0 & (x = 0) \\ x & (x > 0) \end{cases} \qquad \begin{aligned} 2x + y &= 5 \\ x - 3y &= -1 \end{aligned}$$

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5. Famous Mathematical & Physical Equations (Samples)

Algebra, Analysis, and Geometry

• Euler's Identity & Euler's Formula $$e^{i\pi} + 1 = 0, \qquad e^{i\theta} = \cos\theta + i\sin\theta$$
• Gaussian Integral $$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$$
• Basel Problem ($\zeta(2)$) $$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \frac{\pi^2}{6}$$
• Cauchy's Integral Formula $$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z - a} \, dz$$
• Fourier Transform & Inverse Fourier Transform $$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx, \qquad f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i x \xi} \, d\xi$$
• Binomial Theorem $$(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$
• Cauchy-Schwarz Inequality $$\left( \sum_{k=1}^n a_k b_k \right)^2 \le \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

Physics and Applied Mathematics

• Maxwell's Equations (Electromagnetism) $$\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \qquad & \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$
• Schrödinger Equation (Quantum Mechanics) $$i\hbar \frac{\partial}{\partial t} \psi(\mathbf{r}, t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}, t) \right) \psi(\mathbf{r}, t)$$
• Einstein Field Equations (General Relativity) $$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$
• Navier-Stokes Equations (Fluid Dynamics) $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$
• Heat Equation / Diffusion Equation $$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$$
• Normal Distribution (Probability Density Function) $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(x - \mu)^2}{2\sigma^2} \right)$$