\documentclass[a4paper]{article}
\usepackage{xcolor}
\begin{document}
\begin{eqnarray*}
f (z)
 & = & \frac{1}{g (z)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (z)
 & = & \frac{1}{g (z)} \\
& = & \frac{1}{s + it} \\
& = & \frac{1}{s + it} \frac{s - it}{s - it} \\
& = & \frac{s - it}{s^{2} + t^{2}} \\
& = & \underbrace{\frac{s}{s^{2} + t^{2}}}_{u} + i \underbrace{\frac{- t}{s^{2} + t^{2}}}_{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x}
 & = & \frac{\partial u}{\partial s} \frac{\partial s}{\partial x}
 + \frac{\partial u}{\partial t} \frac{\partial t}{\partial x} \\
& = & \frac{\partial}{\partial s} \left ( \frac{s}{s^{2} + t^{2}} \right ) \frac{\partial s}{\partial x}
 + \frac{\partial}{\partial t} \left ( \frac{s}{s^{2} + t^{2}} \right ) \frac{\partial t}{\partial x} \\
& = & \frac{1 \cdot (s^{2} + t^{2}) - s (2s)}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- s}{(s^{2} + t^{2})^{2}} (2t) \frac{\partial t}{\partial x} \\
& = & \frac{s^{2} + t^{2} - 2s^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
& = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y}
 & = & \frac{\partial u}{\partial s} \frac{\partial s}{\partial y}
 + \frac{\partial u}{\partial t} \frac{\partial t}{\partial y} \\
& = & \frac{\partial}{\partial s} \left ( \frac{s}{s^{2} + t^{2}} \right ) \frac{\partial s}{\partial y}
 + \frac{\partial}{\partial t} \left ( \frac{s}{s^{2} + t^{2}} \right ) \frac{\partial t}{\partial y} \\
& = & \frac{1 \cdot (s^{2} + t^{2}) - s (2s)}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- s}{(s^{2} + t^{2})^{2}} (2t) \frac{\partial t}{\partial y} \\
& = & \frac{s^{2} + t^{2} - 2s^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
& = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial x}
 & = & \frac{\partial v}{\partial s} \frac{\partial s}{\partial x}
 + \frac{\partial v}{\partial t} \frac{\partial t}{\partial x} \\
& = & \frac{\partial}{\partial s} \left ( \frac{- t}{s^{2} + t^{2}} \right ) \frac{\partial s}{\partial x}
 + \frac{\partial}{\partial t} \left ( \frac{- t}{s^{2} + t^{2}} \right ) \frac{\partial t}{\partial x} \\
& = & \frac{t}{(s^{2} + t^{2})^{2}} (2s) \frac{\partial s}{\partial x}
 + \frac{(-1) (s^{2} + t^{2}) - (-t) (2t)}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
& = & \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- s^{2} - t^{2} + 2t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
& = & \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial y}
 & = & \frac{\partial v}{\partial s} \frac{\partial s}{\partial y}
 + \frac{\partial v}{\partial t} \frac{\partial t}{\partial y} \\
& = & \frac{\partial}{\partial s} \left ( \frac{- t}{s^{2} + t^{2}} \right ) \frac{\partial s}{\partial y}
 + \frac{\partial}{\partial t} \left ( \frac{- t}{s^{2} + t^{2}} \right ) \frac{\partial t}{\partial y} \\
& = & \frac{t}{(s^{2} + t^{2})^{2}} (2s) \frac{\partial s}{\partial y}
 + \frac{(-1) (s^{2} + t^{2}) - (-t) (2t)}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
& = & \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- s^{2} - t^{2} + 2t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
& = & \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
\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{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x}
 - \left ( \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \right ) \\
& = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x}
 - \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 - \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y} \\
& = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}}
 \underbrace{\left ( \frac{\partial s}{\partial x} - \frac{\partial t}{\partial y} \right )}_{\mbox{CR1 van } g}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}}
 \underbrace{\left ( \frac{\partial t}{\partial x} + \frac{\partial s}{\partial y} \right )}_{\mbox{CR2 van } g} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}
 & = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial y}
 + \frac{- 2st}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial y}
 + \frac{2st}{(s^{2} + t^{2})^{2}} \frac{\partial s}{\partial x}
 + \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}} \frac{\partial t}{\partial x} \\
& = & \frac{- s^{2} + t^{2}}{(s^{2} + t^{2})^{2}}
 \underbrace{\left ( \frac{\partial s}{\partial y} + \frac{\partial t}{\partial x} \right )}_{\mbox{CR2 van } g}
 + \frac{2st}{(s^{2} + t^{2})^{2}}
 \underbrace{\left ( \frac{\partial s}{\partial x} - \frac{\partial t}{\partial y} \right )}_{\mbox{CR1 van } g} \\
\end{eqnarray*}
\newpage
\color{violet}
\begin{eqnarray*}
f (z)
 & = & \frac{1}{g (z)} \\
\end{eqnarray*}
\end{document}
