\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (z)
 & = & \ln | \, z \, | + i \arg z \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (z)
 & = & \ln | \, z \, | + i \arg z \\
& = & \ln \sqrt{x^{2} + y^{2}} + i \arctan \frac{y}{x} \\
& = & \underbrace{\frac{1}{2} \ln (x^{2} + y^{2})}_{u} + i \underbrace{\arctan \frac{y}{x}}_{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x}
 & = & \frac{\partial}{\partial x} \left ( \frac{1}{2} \ln (x^{2} + y^{2}) \right ) \\
& = & \frac{1}{2} \frac{1}{x^{2} + y^{2}} (2x) \\
& = & \frac{x}{x^{2} + y^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y}
 & = & \frac{\partial}{\partial y} \left ( \frac{1}{2} \ln (x^{2} + y^{2}) \right ) \\
& = & \frac{1}{2} \frac{1}{x^{2} + y^{2}} (2y) \\
& = & \frac{y}{x^{2} + y^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial x}
 & = & \frac{\partial}{\partial x} \left ( \arctan \frac{y}{x} \right ) \\
& = & \frac{1}{1 + \left ( \frac{y}{x} \right )^{2}} \left ( - \frac{y}{x^{2}} \right ) \\
& = & \frac{1}{1 + \frac{y^{2}}{x^{2}}} \left ( - \frac{y}{x^{2}} \right ) \\
& = & - \frac{y}{x^{2} + y^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial y}
 & = & \frac{\partial}{\partial y} \left ( \arctan \frac{y}{x} \right ) \\
& = & \frac{1}{1 + \left ( \frac{y}{x} \right )^{2}} \left ( \frac{1}{x} \right ) \\
& = & \frac{1}{1 + \frac{y^{2}}{x^{2}}} \left ( \frac{x}{x^{2}} \right ) \\
& = & \frac{x}{x^{2} + y^{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}{x^{2} + y^{2}} - \frac{x}{x^{2} + y^{2}} \\
& = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}
 & = & \frac{y}{x^{2} + y^{2}} + \left ( - \frac{y}{x^{2} + y^{2}} \right ) \\
& = & \frac{y}{x^{2} + y^{2}} - \frac{y}{x^{2} + y^{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}{x^{2} + y^{2}} + i \left ( - \frac{y}{x^{2} + y^{2}} \right ) \\
& = & \frac{x}{x^{2} + y^{2}} - i \frac{y}{x^{2} + y^{2}} \\
& = & \frac{x - iy}{x^{2} + y^{2}} \\
& = & \frac{x - iy}{(x + iy) (x - iy)} \\
& = & \frac{1}{x + iy} \\
& = & \frac{1}{z} \\
\end{eqnarray*}
\end{document}
