# Hello MathJax!!

The following equations are represented in the HTML source code as LaTeX expressions.
The Cauchy-Schwarz Inequality
$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$
Auto Numbering and Ref?
Suppose that $f$ is a function with period $\pi$, then we have
$\newcommand{\rd}{\rm d}$
\begin{gather}\label{eq:1}
\int_t^{t+\pi} f(x)\rd x=\int_0^\pi f(x)\rd x.\tag{1}
\end{gather}
From \eqref{eq:1} we know that $\dots$.
Can we use \newcommand to define some macro?
$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$
define newcommand as:

We consider, for various values of $s$, the $n$-dimensional integral
\begin{align}\label{def:Wns}
W_n (s):=
\int_{[0, 1]^n}
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
\end{align}
which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.

By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer

\begin{align}\label{eq:W3k}
W_3(k)= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align}
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
Definition of Christoffel Symbols

$(∇XY)k=Xi(∇iY)k=Xi(∂Yk∂xi+ΓimkYm)$

1. awj141

for example,
the roots of the real polynomial $f(x)=ax^2+bx+c$ is
$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

• 魏美春

有创意

• awj141

呵呵, 就是想学学怎么在网页插入tex