Math Formula Test

This is a test of math formula rendering using KaTeX.

Inline Math

The inline math formula: $E = mc^2$

Another inline formula: $\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$

The block math formula:

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

The Schrödinger equation:

$$ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \left [ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right ] \Psi(\mathbf{r},t) $$

The Cauchy-Schwarz inequality:

$$ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) $$

Inline math in text: When $a \neq 0$, there are two solutions to $ax^2 + bx + c = 0$ and they are $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.