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\begin{document}

\vspace*{-2cm}

\centerline{\Large\textbf{Escritura de fórmulas matemáticas: Ejemplos}}

\vspace{1.5cm}

\textbf{1.-} \ \
\begin{equation*}
\biggl(\prod^n_{\,j=1}\hat x_j\biggr)H_c=\frac{1}{2}\hat k_{ij}\det
\widehat{\mathbf{K}}(i|i),\qquad i=1,\dots,n.
\end{equation*}

\bigskip

\textbf{2.-} \ \
\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}
{\dbar{v}} = 0 \]

\bigskip

\textbf{3.-} \ \
\begin{equation*}
H_c= \frac{n_1!\,n_2!\,n_3!}
{n_1+n_2+n_3}\sum_i\left[\binom{n_1}{i}
\binom{n_2}{n_3-n_1+i}\binom{n_3}{n_3-n_2+i}\right]
\end{equation*}

\bigskip

\textbf{4.-} \ \
\begin{equation*}
\mathbf{K}(t,t_1,\dots,t_n)
=\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
\hdotsfor[2]{4}\\
-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
\end{equation*}

\bigskip

\textbf{5.-} \ \
\begin{equation*}
\int_{\mathcal{D}}\sbar{\overline\partial u}^2\Phi_ 0(z) e^{\alpha\sbar{z}^2}
\geq \alpha\int_{\mathcal{D}}\sbar{u}^2\Phi_ 0e^{\alpha\sbar{z}^2}
\end{equation*}

\bigskip

\textbf{6.-} \ \ \ \ Demostrar que si definimos $L=\lim_{h\to 0^+}(g(hz)-g(0))/h$,
entonces se verifica la equivalencia:
$$\lim_{h\to 0^+}g(\omega(h))=L\Leftrightarrow\lim_{h\to
0^+}g(h)=L$$

\bigskip

\textbf{7.-} \ \
\begin{equation*}
0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1]
\xrightarrow{\partial_0\alpha(b)} E^{\partial_0b}\
\end{equation*}

\bigskip

\textbf{8.-} \ \ Definir un comando \verb+\laplaciano{f}+, dependiente de un argumento, 
que automatice la escritura de expresiones como: 
$$\Delta f = \laplaciano{f}$$
$$\Delta g = \laplaciano{g}$$





\end{document}


