WELCOME TO MATH TESTING.

Formula display

Formula:

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1 2	$T(n) = \Theta(n)$ $$T(n) = \Theta(n)$$

$T(n) = \Theta(n)$

\[T(n) = \Theta(n)\]

1 2	$$R{m \times n} = U{m \times m} S{m \times n} V{n \times n}’$$ $$R_{m \times n} = U_{m \times m} S_{m \times n} V_{n \times n}'$$

\[R{m \times n} = U{m \times m} S{m \times n} V{n \times n}’\] \[R_{m \times n} = U_{m \times m} S_{m \times n} V_{n \times n}'\]

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$$d_{1}=\frac{\ln(S_{t}/K)  +(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$$

$$d\_{1}=\frac{\ln(S\_{t}/K)  +(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$$

\[d_{1}=\frac{\ln(S_{t}/K) +(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}\]

\[d\_{1}=\frac{\ln(S\_{t}/K) +(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}\]

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$$T(n) = \Theta(n) + \sum{i=0}^{n-1}{O({n}{i}^2)}$$
$$T(n) = \Theta(n) + \sum_{i=0}^{n-1}{O({n}_{i}^2)}$$
$$T(n) = \Theta(n) + \sum\_{i=0}^{n-1}{O({n}\_{i}^2)}$$

\[T(n) = \Theta(n) + \sum{i=0}^{n-1}{O({n}{i}^2)}\] \[T(n) = \Theta(n) + \sum_{i=0}^{n-1}{O({n}_{i}^2)}\] \[T(n) = \Theta(n) + \sum\_{i=0}^{n-1}{O({n}\_{i}^2)}\]

Repeating fractions:

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$$ 
frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } } 
$$

\[ frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } } \]

Summation notation:

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

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

Greek Letters:

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$$ 
\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi 
$$

\[ \ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi \]

Symbols:

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1 2	$$ \surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup $$

\[ \surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup \]

Calculus:

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$$
f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}
$$

\[ f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x} \]

Cross Product:

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$$
\mathbf{V}_1 \times \mathbf{V}_2 =  \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} &  \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} &  \frac{\partial Y}{\partial v} & 0
\end{vmatrix}
$$

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix} \]

Matrices:

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$$
\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\ 
a_{21} & a_{22} & a_{23}\\ 
a_{31} & a_{32} & a_{33}
\end{pmatrix}
$$

\[ \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

Long Formula display

$$
\begin{align}
\pi^{-z}\zeta(2z)\Gamma(z)&=\int_0^{+\infty}\left(\sum_{n=1}^\infty e^{-\pi n^2s}\right)s^{z-1}ds\\\\
&=\int_0^{+\infty}\psi(s)s^{z-1}ds
\end{align}
$$

Thus

$$
\begin{align}
\pi^{-z}\zeta(2z)\Gamma(z)&=\int_0^{+\infty}\psi(s)s^{z-1}ds\\\\
&=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_0^1\psi(s)s^{z-1}ds\\\\
&=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_1^{+\infty}\psi\left(\frac{1}{s}\right)s^{1-z}\frac{ds}{s^2}\\\\
&=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_1^{+\infty}\left(\psi(s)\sqrt{s}+\frac{\sqrt{s}-1}{2}\right)s^{-1-z}ds\\\\
&=\int_1^{+\infty}\psi(s)(s^{z-1}+s^{-z-\frac{1}{2}})ds+\int_1^{+\infty}\frac{s^{-z-\frac{1}{2}}-s^{-z-1}}{2}ds\\\\
&=\int_1^{+\infty}\psi(s)(s^{z-1}+s^{-z-\frac{1}{2}})ds+\frac{1}{2z(2z-1)}
\end{align}
$$

\[ \begin{align} \pi^{-z}\zeta(2z)\Gamma(z)&=\int_0^{+\infty}\left(\sum_{n=1}^\infty e^{-\pi n^2s}\right)s^{z-1}ds\\\\ &=\int_0^{+\infty}\psi(s)s^{z-1}ds \end{align} \]

Thus

\[ \begin{align} \pi^{-z}\zeta(2z)\Gamma(z)&=\int_0^{+\infty}\psi(s)s^{z-1}ds\\\\ &=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_0^1\psi(s)s^{z-1}ds\\\\ &=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_1^{+\infty}\psi\left(\frac{1}{s}\right)s^{1-z}\frac{ds}{s^2}\\\\ &=\int_1^{+\infty}\psi(s)s^{z-1}ds+\int_1^{+\infty}\left(\psi(s)\sqrt{s}+\frac{\sqrt{s}-1}{2}\right)s^{-1-z}ds\\\\ &=\int_1^{+\infty}\psi(s)(s^{z-1}+s^{-z-\frac{1}{2}})ds+\int_1^{+\infty}\frac{s^{-z-\frac{1}{2}}-s^{-z-1}}{2}ds\\\\ &=\int_1^{+\infty}\psi(s)(s^{z-1}+s^{-z-\frac{1}{2}})ds+\frac{1}{2z(2z-1)} \end{align} \]

Inline formula

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This is an example for $x_mu$ and $y_mu$. This is an example for $x_{mu}$ and $y_{mu}$. This is an example for $x\_mu$ and $y\_mu$. $R{m \times n} = U{m \times m} S{m \times n} V{n \times n}’$，$d_{1}=\frac{\ln(S_{t}/K)+(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$。

This is an example for $x_mu$ and $y_mu$. This is an example for $x_{mu}$ and $y_{mu}$. This is an example for $x\_mu$ and $y\_mu$. $R{m \times n} = U{m \times m} S{m \times n} V{n \times n}’$，$d_{1}=\frac{\ln(S_{t}/K)+(r+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$。

Numbering and referring equations

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The Maxwell's Equations:

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$$
\begin{eqnarray}
\nabla\cdot\vec{E} &=& \frac{\rho}{\epsilon_0} \\
\nabla\cdot\vec{B} &=& 0 \\
\nabla\times\vec{E} &=& -\frac{\partial B}{\partial t} \\
\nabla\times\vec{B} &=& \mu_0\left(\vec{J}+\epsilon_0\frac{\partial E}{\partial t} \right)
\end{eqnarray}
$$

\[ \begin{eqnarray} \nabla\cdot\vec{E} &=& \frac{\rho}{\epsilon_0} \\ \nabla\cdot\vec{B} &=& 0 \\ \nabla\times\vec{E} &=& -\frac{\partial B}{\partial t} \\ \nabla\times\vec{B} &=& \mu_0\left(\vec{J}+\epsilon_0\frac{\partial E}{\partial t} \right) \end{eqnarray} \]

The famous matter-energy equation $\ref{eq1}$ proposed by Einstein ...

$$
\begin{equation}
\label{eq1}
e=mc^2
\end{equation}
$$

\[ \begin{equation} \label{eq1} e=mc^2 \end{equation} \]

For multi-line equations shown in Eq. $\eqref{eq2}$, inside the equation environment, you can use the aligned environment to split it into multiple lines:

$$
\begin{equation}
\begin{aligned}
a &= b + c \\
  &= d + e + f + g \\
  &= h + i
\end{aligned}
\end{equation}
\label{eq2}
$$

\[ \begin{equation} \begin{aligned} a &= b + c \\ &= d + e + f + g \\ &= h + i \end{aligned} \end{equation} \label{eq2} \]

We can use align environment to align multiple equations shown in Eq. $\eqref{eq5}$. Each of these equations will get its own numbers.

$$
\begin{align}
a &= b + c \label{eq3} \\
x &= yz \label{eq4}\\
l &= m - n 
\label{eq5}
\end{align}
$$

\[ \begin{align} a &= b + c \label{eq3} \\ x &= yz \label{eq4}\\ l &= m - n \label{eq5} \end{align} \]

In the align environment, if you do not want to number one or some equations shown in Eq. $\eqref{eq6}$, just use \nonumber right behind these equations. Like the following:

$$
\begin{align}
-4 + 5x &= 2+y \nonumber  \\
 w+2 &= -1+w \label{eq6} \\
 ab &= cb
\end{align}
$$

\[ \begin{align} -4 + 5x &= 2+y \nonumber \\ w+2 &= -1+w \label{eq6} \\ ab &= cb \end{align} \]

Sometimes, you want to use more “exotic” style to refer your equation shown in Eq. $\eqref{eq_tag}$. You can use \tag{} to achieve this. For example:

$$
x+1\over\sqrt{1-x^2} 
\tag{i}
\label{eq_tag}
$$

\[ x+1\over\sqrt{1-x^2} \tag{i} \label{eq_tag} \]