\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (z)
 & = & \frac{1}{z - z_{0}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (z)
 & = & \frac{1}{z - z_{0}} \\
& = & \frac{1}{x + yi - (a + bi)} \\
& = & \frac{1}{x - a + i (y - b)} \\
& = & \frac{1}{x - a + i (y - b)} \frac{x - a - i (y - b)}{x - a - i (y - b)} \\
& = & \frac{x - a - i (y - b)}{(x - a)^{2} + (y - b)^{2}} \\
& = & \underbrace{\frac{x - a}{(x - a)^{2} + (y - b)^{2}}}_{u} + i \underbrace{\frac{- (y - b)}{(x - a)^{2} + (y - b)^{2}}}_{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x}
 & = & \frac{\partial}{\partial x} \left ( \frac{x - a}{(x - a)^{2} + (y - b)^{2}} \right ) \\
& = & \frac{(1) ((x - a)^{2} + (y - b)^{2}) - (x - a) (2 (x - a))}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{(x - a)^{2} + (y - b)^{2} - 2 (x - a)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{- (x - a)^{2} + (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y}
 & = & \frac{\partial}{\partial y} \left ( \frac{x - a}{(x - a)^{2} + (y - b)^{2}} \right ) \\
& = & \frac{(0) ((x - a)^{2} + (y - b)^{2}) - (x - a) (2 (y - b))}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{- 2 (x - a) (y - b)}{((x - a)^{2} + (y - b)^{2})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial x}
 & = & \frac{\partial}{\partial x} \left ( \frac{- (y - b)}{(x - a)^{2} + (y - b)^{2}} \right ) \\
& = & \frac{(0) ((x - a)^{2} + (y - b)^{2}) + (y - b) (2 (x - a))}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{2 (x - a) (y - b)}{((x - a)^{2} + (y - b)^{2})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial y}
 & = & \frac{\partial}{\partial y} \left ( \frac{- (y - b)}{(x - a)^{2} + (y - b)^{2}} \right ) \\
& = & \frac{(- 1) ((x - a)^{2} + (y - b)^{2}) + (y - b) (2 (y - b))}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{- (x - a)^{2} - (y - b)^{2} + 2 (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & \frac{- (x - a)^{2} + (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}
 & = & 0 \quad \wedge \quad \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \, \, \, = \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}
 & = & \frac{- (x - a)^{2} + (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} - \frac{- (x - a)^{2} + (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}
 & = & \frac{- 2 (x - a) (y - b)}{((x - a)^{2} + (y - b)^{2})^{2}} + \frac{2 (x - a) (y - b)}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\mbox{d}f (z)}{\mbox{d}z}
 & = & \frac{\partial u}{\partial x} + i \, \frac{\partial v}{\partial x} \\
& = & \frac{\partial v}{\partial y} - i \, \frac{\partial u}{\partial y} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\mbox{d}f (z)}{\mbox{d}z}
 & = & \frac{\partial u}{\partial x} + i \, \frac{\partial v}{\partial x} \\
& = & \frac{- (x - a)^{2} + (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} + i \frac{2 (x - a) (y - b)}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & - \frac{(x - a)^{2} - 2 i (x - a) (y - b) - (y - b)^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & - \frac{((x - a) - i (y - b))^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \\
& = & - \frac{((x - a) - i (y - b))^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \frac{((x - a) + i (y - b))^{2}}{((x - a) + i (y - b))^{2}} \\
& = & - \frac{((x - a)^{2} + (y - b)^{2})^{2}}{((x - a)^{2} + (y - b)^{2})^{2}} \frac{1}{((x - a) + i (y - b))^{2}} \\
& = & - \frac{1}{((x - a) + i (y - b))^{2}} \\
& = & - \frac{1}{(x - a + i (y - b))^{2}} \\
& = & - \frac{1}{(x + yi - (a + bi))^{2}} \\
& = & - \frac{1}{(z - z_{0})^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{Noemer}
 & = & 0 \\
z - z_{0}
 & = & 0 \\
x + yi - (a + bi)
 & = & 0 \\
x + yi - a - bi
 & = & 0 \\
x
 & = & a \quad \wedge \quad y \, \, \, = \, \, \, b \\
\end{eqnarray*}
\end{document}
