test_formula

Here are various LaTeX formulas to test rendering capabilities:

Einstein’s famous equation: $E = mc^2$ shows the relationship between energy and mass.

The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves $ax^2 + bx + c = 0$ .

The Pythagorean theorem states that $a^2 + b^2 = c^2$ for right triangles.

Newton’s second law of motion:

F = m \cdot a

Euler’s identity, often considered the most beautiful equation in mathematics:

e^{i\pi} + 1 = 0

The Gaussian integral:

\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

A matrix representation:

\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}

A system of linear equations:

\begin{aligned} 3x + 2y - z &= 1 \\ 2x - 2y + 4z &= -2 \\ -x + \frac{1}{2}y - z &= 0 \end{aligned}

A piecewise function definition:

\begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases}

The expectation value in quantum mechanics:

\left\langle \psi \right| \hat{A} \left| \psi \right\rangle

A binomial coefficient:

\binom{n}{k} = \left( \begin{array}{c} n \\ k \end{array} \right) = \frac{n!}{k!(n-k)!}

The Legendre symbol:

\left( \frac{a}{p} \right) = \begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p \text{ and } a \not\equiv 0 \pmod{p} \\ -1 & \text{if } a \text{ is a quadratic non-residue modulo } p \\ 0 & \text{if } a \equiv 0 \pmod{p} \end{cases}

The Fourier transform:

\mathcal{F}[f(t)] = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt

Schrödinger’s equation:

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)

Maxwell’s equations in differential form:

\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} \\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t} \end{aligned}

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О.К.Б Саблин