=for timestamp
Mi Jun 15 18:44:36 CEST 2005

=head3 25. Hausaufgabe

=head4 Selbstgestellte Aufgabe

M<z \mapsto w = -2\mathrm{i}z^* + 2 + 4\mathrm{i};>

M<x_0 + \mathrm{i}y_0 = -2\mathrm{i}\left(x_0 - \mathrm{i}y_0\right) + 2 +
4\mathrm{i} = -2\mathrm{i}x_0 - 2y_0 + 2 + 4\mathrm{i};>

⇒ M<x_0 + 2y_0 - 2 = \mathrm{i}\left(-2x_0 - y_0 + 4\right);>

⇒ M<<
{} \left\{\begin{array}{rcl}
{}     x_0 + 2y_0 - 2 &=& 0; \Rightarrow x_0 = 2 - 2y_0; \\
{}   -2x_0 -  y_0 + 4 &=& 0; \Rightarrow x_0 = 2 - \frac{y_0}{2}; \quad y_0 = 4 - 2x_0;
{} \end{array} \right\} \\\Rightarrow
{} \left\{\begin{array}{l}
{}   2 - 2y_0 = 2 - \frac{y_0}{2}; \Rightarrow 4y_0 = y_0; \Rightarrow y_0 = 0; \\
{}   x_0 = 2 - 2\left(4 - 2x_0\right) = 2 - 8 + 4x_0 = -6 + 4x_0 = 0;
\Rightarrow x_0 = 2;
{} \end{array}\right.
>>

⇒ M<z_0 = 2;>

=helper MyBook::Helper::Gnuplot

# Global settings
set samples 10000
unset border
set xtics axis
set ytics axis
set xzeroaxis lt -1
set yzeroaxis lt -1

# Coordinate system settings
set title ""
set xlabel "\nRe(w)\n"
set ylabel "Im(w)\n"
set xrange [ -7.000000 :  5.000000 ]
set yrange [ -1.000000 : 13.000000 ]
set grid
set xtics 2.000000
set ytics 2.000000

plot \
  "<echo -e '0 0\\n0 4\\n-4 4\\n-4 0\\n0 0'" t "z" w lp, \
  "<echo -e '2 4\\n-6 4\\n-6 12\\n2 12\\n2 4'" t "w" w lp

=hend