=for timestamp
Mi Apr 13 16:10:15 CEST 2005

=head3 18. Hausaufgabe

=head4 Selbstgestellte Aufgabe

M<\mathrm{f}(z) = az + b;>

M<z_1 = 3\mathrm{i} \mapsto \mathrm{f}(z_1) = -2 - 4\mathrm{i}; \\>
M<z_2 = 0           \mapsto \mathrm{f}(z_2) = -2 - \mathrm{i};  \\>
M<z_3 = 2           \mapsto \mathrm{f}(z_3);>

M<3\mathrm{i}a + b = -2 - 4\mathrm{i}; \Rightarrow b = -2 - 4\mathrm{i} -
3\mathrm{i}a;>

M<b = -2 - \mathrm{i};>

⇒ M<-2 - 4\mathrm{i} - 3\mathrm{i}a = -2 - \mathrm{i}; \Rightarrow a = -1;>

⇒ M<\mathrm{f}(z_0) = -z_0 - 2 - i = z_0; \Rightarrow z_0 = -1 - \frac{\mathrm{i}}{2};>

⇒ M<\mathrm{f}(z_3) = -4 - \mathrm{i};>

=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 "\n\nRe(z), Re(f(z))\n"
set ylabel "Im(z), Im(f(z))\n\n"
set xrange [ -5 : 3 ]
set yrange [ -5 : 4 ]
set grid
set xtics 2.000000
set ytics 2.000000

plot \
  "<echo -e ' 0  3\\n 0  0\\n 2  0\\n 0  3'" t "z" w lp, \
  "<echo -e '-2 -4\\n-2 -1\\n-4 -1\\n-2 -4'" t "f(z)" w lp, \
  "<echo -e '-1 -0.5'" t "Fixpunkt" w p

=hend