complex1.tex

\begin{array}{cc}
\mathrm{Bernoulli\ Trials} & {P}({E})=\left( \genfrac{}{}{0.0mm}{}{{n}}{{k}} \right) {{p}}_{}^{{k}}{(1-{p})}_{}^{{n}-{k}} \\
\mathrm{Cauchy-Schwarz\ Inequality} & {\left( \underset{{k}=1}{\overset{{n}}{\sum }}{{a}}_{{k}}^{}{{b}}_{{k}}^{}\right) }_{}^{2}\leq \left( \underset{{k}=1}{\overset{{n}}{\sum }}{{a}}_{{k}}^{2}\right) \left( \underset{{k}=1}{\overset{{n}}{\sum }}{{b}}_{{k}}^{2}\right)  \\
\mathrm{Cauchy\ Formula} & {f}({z})\mathrm{\,\ }{{Ind}}_{{\gamma }}^{}({z})=\frac{1}{2{\pi }{i}} \underset{{\gamma }}{\overset{}{\oint }}\frac{{f}({\xi })}{{\xi }-{z}} \mathrm{\,\ }{d}{\xi } \\
\mathrm{Cross\ Product} & {{V}}_{1}^{}\times {{V}}_{2}^{}=\left| \begin{array}{ccc}
{i} & {j} & {k} \\
\frac{\partial {X}}{\partial {u}}  & \frac{\partial {Y}}{\partial {u}}  & 0 \\
\frac{\partial {X}}{\partial {v}}  & \frac{\partial {Y}}{\partial {v}}  & 0 \\
\end{array}\right|  \\
\mathrm{Vandermonde\ Determinant} & \left| \begin{array}{cccc}
1 & 1 & \cdots  & 1 \\
{{v}}_{1}^{} & {{v}}_{2}^{} & \cdots  & {{v}}_{{n}}^{} \\
{{v}}_{1}^{2} & {{v}}_{2}^{2} & \cdots  & {{v}}_{{n}}^{2} \\
\vdots  & \vdots  & \ddots  & \vdots  \\
{{v}}_{1}^{{n}-1} & {{v}}_{2}^{{n}-1} & \cdots  & {{v}}_{{n}}^{{n}-1} \\
\end{array}\right| =\underset{1\leq {i}<{j}\leq {n}}{\overset{}{\prod }}({{v}}_{{j}}^{}-{{v}}_{{i}}^{}) \\
\mathrm{Lorenz\ Equations} & \begin{array}{ccc}
\overset{}{\underset{}{{x}}}  & = & {\sigma }({y}-{x}) \\
\overset{}{\underset{}{{y}}}  & = & {\rho }{x}-{y}-{x}{z} \\
\overset{}{\underset{}{{z}}}  & = & -{\beta }{z}+{x}{y} \\
\end{array} \\
\mathrm{Maxwell's\ Equations} & \left\{  \begin{array}{ccc}
\nabla \mathrm{}\times \overset{\leftharpoonup }{\underset{}{{B}}} -\mathrm{\,\ }\frac{1}{{c}} \mathrm{\,\ }\frac{\partial \mathrm{}\overset{\leftharpoonup }{\underset{}{{E}}} }{\partial \mathrm{}{t}}  & = & \frac{4{\pi }}{{c}} \mathrm{\,\ }\overset{\leftharpoonup }{\underset{}{{j}}}  \\
\nabla \mathrm{}\overset{\leftharpoonup }{\underset{}{{E}}}  & = & 4{\pi }{\rho } \\
\nabla \mathrm{}\times \overset{\leftharpoonup }{\underset{}{{E}}} \mathrm{\,\ }+\mathrm{\,\ }\frac{1}{{c}} \mathrm{\,\ }\frac{\partial \mathrm{}\overset{\leftharpoonup }{\underset{}{{B}}} }{\partial \mathrm{}{t}}  & = & \overset{\leftharpoonup }{\underset{}{0}}  \\
\nabla \mathrm{}\overset{\leftharpoonup }{\underset{}{{B}}}  & = & 0 \\
\end{array}\right.  \\
\mathrm{Einstein\ Field\ Equations} & {{R}}_{{\mu }{\nu }}^{}-\frac{1}{2} \mathrm{\,\ }{{g}}_{{\mu }{\nu }}^{}\mathrm{\,\ }{R}=\frac{8{\pi }{G}}{{{c}}_{}^{4}} \mathrm{\,\ }{{T}}_{{\mu }{\nu }}^{} \\
\mathrm{Ramanujan\ Identity} & \frac{1}{(\sqrt{{\varphi }\sqrt{5}}-{\varphi }){{e}}_{}^{\frac{25}{{\pi }} }} =1+\frac{{{e}}_{}^{-2{\pi }}}{1+\frac{{{e}}_{}^{-4{\pi }}}{1+\frac{{{e}}_{}^{-6{\pi }}}{1+\frac{{{e}}_{}^{-8{\pi }}}{1+\ldots } } } }  \\
\mathrm{Another\ Ramanujan\ identity} & \underset{{k}=1}{\overset{{\infty }}{\sum }}\frac{1}{{2}_{}^{\lfloor {k}\mathrm{}{\varphi }\rfloor }} =\frac{1}{{2}_{}^{0}+\frac{1}{{2}_{}^{1}+\cdots } }  \\
\mathrm{Rogers-Ramanujan\ Identity} & 1+\underset{{k}=1}{\overset{{\infty }}{\sum }}\frac{{{q}}_{}^{{{k}}_{}^{2}+{k}}}{(1-{q})(1-{{q}}_{}^{2})\cdots (1-{{q}}_{}^{{k}})} =\underset{{j}=0}{\overset{{\infty }}{\prod }}\frac{1}{(1-{{q}}_{}^{5{j}+2})(1-{{q}}_{}^{5{j}+3})} ,\mathrm{\:\ }\mathrm{\:\ }{f}{o}{r}\mathrm{\:\ }|{q}|<1{.} \\
\mathrm{Commutative\ Diagram} & \begin{array}{ccc}
{H} & \leftarrow  & {K} \\
\downarrow  &  \qquad & \uparrow  \\
{H} & \rightarrow  & {K} \\
\end{array} \\
\end{array}