Testing LaTeX equations

This document demonstrates how to use $\LaTeX$ equations in your Markdown files for AstroPaper. $\LaTeX$ is a powerful typesetting system often used for mathematical and scientific documents.

Inline Equations

Inline equations are written between single dollar signs $...$. Here are some examples:

1. The famous mass-energy equivalence formula $E = mc^2$ : $E = mc^2$
2. The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ : $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
3. Euler’s identity $e^{i\pi} + 1 = 0$ : $e^{i\pi} + 1 = 0$

Block Equations

For more complex equations or when you want the equation to be displayed on its own line, use double dollar signs $$...$$:

The Gaussian integral

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

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

The definition of the Riemann zeta function

$$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$

Maxwell’s equations in differential form

$$
\begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)
\end{aligned}
$$

Using Mathematical Symbols

LaTeX provides a wide range of mathematical symbols:

• Greek letters: $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\pi$
• Operators: $\sum$, $\prod$, $\int$, $\partial$, $\nabla$
• Relations: $\leq$, $\geq$, $\approx$, $\sim$, $\propto$
• Logical symbols: $\forall$, $\exists$, $\neg$, $\wedge$, $\vee$

Testing specific characters:

$H_a: \theta>\theta_0$

Results:

$H_a: \theta>\theta_0$

$\not\in$
$\neq$
$\neg$
$\sim$
$\nexists$
$\lambda$
$\lmoustache\0\2pi$

Results

$\not\in$ $\neq$ $\neg$ $\sim$ $\nexists$ $\lambda$ $\lmoustache\pi$

$$
\begin{equation*}
\begin{aligned}
\iiint_{\mathcal{Q}} f(w,x,y,z) \,dw \,dx \,dy \,dz
&\leq 
\oint_{\partial Q} f'
  \left(
    \max \left\{ \frac{|w|}{|{w^2 + x^2}|} ; 
    \frac{|z|}{|{y^2 + z^2}|} ;
    \frac{|{w \oplus z}|}{|{x \oplus y}|} \right\}
  \right)
\\
&\precapprox
\biguplus_{\mathbb{Q} \Subset \bar{Q}}
  \left[ f^* \left(
    \frac{\lmoustache \mathbb{Q}(t)\rmoustache}{\sqrt{1 - t^2}}
  \right) \right]^{t=9}_{t=\alpha}
\end{aligned}
\end{equation*}
$$

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